Question: Which Grade I listed building in Dorset, England was designed by the same designer as Blenheim Palace and Castle Howard? 

Context articles: Courtyard house
Easton Neston
John Vanbrugh
Eastbury Park
Grade II* listed war memorials in England
Atlas Fountain
Nicholas Hawksmoor
Listed buildings in Appleby-in-Westmorland
Listed buildings in Frodsham
Grade I listed war memorials in England

A courtyard house is a type of house—often a large house—where the main part of the building is disposed around a central courtyard.
 Many houses that have courtyards are not courtyard houses of the type covered by this article.
 For example, large houses often have small courtyards surrounded by service rooms or corridors, but the main rooms are not disposed around a courtyard.
 Blenheim Palace in England is an example of such a house.
Easton Neston is a large grade I listed country house in the parish of Easton Neston near Towcester in Northamptonshire, England.
 It was built by William Fermor, 1st Baron Leominster (1648-1711), in the Baroque style to the design of the architect Nicholas Hawksmoor.
 Easton Neston is thought to be the only mansion which was solely the work of Hawksmoor.
 From about 1700, after the completion of Easton Neston, Hawksmoor worked with Sir John Vanbrugh on many buildings, including Castle Howard and Blenheim Palace, and often provided technical knowledge to the less qualified Vanbrugh.
 Hawksmoor's work was always more classically severe than Vanbrugh's.
 However, Easton Neston predates this partnership by some six years.
Sir John Vanbrugh ( ; 24 January 1664 (baptised) – 26 March 1726) was an English architect and dramatist, perhaps best known as the designer of Blenheim Palace and Castle Howard.
 He wrote two argumentative and outspoken Restoration comedies, "The Relapse" (1696) and "The Provoked Wife" (1697), which have become enduring stage favourites but originally occasioned much controversy.
 He was knighted in 1714.
Eastbury Park was a country estate near Tarrant Gunville in Dorset, England.
 It contained a large mansion designed by Sir John Vanbrugh.
 The mansion has not survived, but its former service wing has become a country house known as Eastbury House, a Grade I listed building.
There are 61 Grade II* listed war memorials in England, out of over 2,000 listed war memorials.
 In the United Kingdom, a listed building is a building or structure of special historical or architectural importance; listing offers the building legal protection against demolition or modification, which requires permission from the local planning authority.
 Listed buildings are divided into three categories—grade I, grade II*, and grade II—which reflect the relative significance of the structure and may be a factor in planning decisions.
 Grade I is the most significant and accounts for 2.5% of listed buildings, while grade II accounts for 92%.
 Grade II* is the intermediate grade accounting for the remaining 5.5%; it is reserved for "particularly important buildings of more than special interest".
 A war memorial listed at grade II* may be of particular artistic interest or accomplishment, of a highly unusual design, or of significant historical interest below that required for grade I.
 It is explicitly unnecessary for the architect or sculptor to be well known in order for a memorial to be listed at grade II*.
 As part of the commemorations of the centenary of the First World War, Historic England—the government body responsible for listing in England—is running a project with the aim of significantly increasing the number of war memorials on the National Heritage List for England.
The Atlas Fountain stands in the grounds of Castle Howard, North Yorkshire, England.
 It has been designated a Grade I listed building and is now recorded in the National Heritage List for England, maintained by Historic England.
Nicholas Hawksmoor (probably 1661 – 25 March 1736) was an English architect.
 He was a leading figure of the English Baroque style of architecture in the late-seventeenth and early-eighteenth centuries.
 Hawksmoor worked alongside the principal architects of the time, Christopher Wren and John Vanbrugh, and contributed to the design of some of the most notable buildings of the period, including St Paul's Cathedral, Wren's City of London churches, Blenheim Palace and Castle Howard.
 Part of his work has been correctly attributed to him only relatively recently, and his influence has reached several poets and authors of the twentieth century.
Appleby-in-Westmorland is a civil parish in the Eden District, Cumbria, England.
 It contains 140 buildings that are recorded in the National Heritage List for England.
 Of these, six are listed at Grade I, the highest of the three grades, nine are at Grade II*, the middle grade, and the others are at Grade II, the lowest grade.
 The parish contains the market town of Appleby (renamed Appleby-in-Westmorland in 1974) and surrounding countryside.
 There is a great variety of types of listed building in the parish.
 The most important building is Appleby Castle; this and a number of associated structures are listed.
 Most of the listed buildings are houses and associated structures, shops and public buildings.
 Other listed buildings include churches, public houses, hotels, two crosses and a lamp post, farmhouses and farm buildings, former industrial buildings, schools, banks, a bridge, railway station buildings, and a milestone.
Frodsham is a civil parish in Cheshire West and Chester, England.
 It contains 61 buildings that are recorded in the National Heritage List for England as designated listed buildings. Of these, one is listed at Grade I, the highest grade, another one is listed at Grade II*, the middle grade, and the others are at Grade II.
 The parish consists of the market town of Frodsham and surrounding countryside.
 The River Weaver and the Weaver Navigation run through the northern part of the parish, and there are two associated listed structures.
 The Warrington to Chester railway line runs in a north-south direction through the parish.
 Frodsham railway station, and the Frodsham viaduct crossing the Weaver, are listed.
 The oldest listed building, and the only one at Grade I is St Laurence's Church, which contains some Norman features.
 The Grade II* building is the former vicarage, which was extended in the 19th century by John Douglas.
 Most of the other listed buildings are houses and associated structures, shops, and cottages, some of which date back to the 16th and 17th centuries and are timber-framed.
 The other listed structure include farmhouses, farm buildings, public houses, hotels, boundary stones, tide stones, a sundial, memorials, and telephone kiosks.
 The newest listed building is a former anti-aircraft operations room that was built in about 1951.
There are 20 Grade I listed war memorials in England, out of over 3,000 listed war memorials.
 In the United Kingdom, a listed building is a building or structure of special historical or architectural importance; listing offers the building legal protection against demolition or modification, which requires permission from the local planning authority.
 Listed buildings are divided into three categories—grade I, grade II*, and grade II—which reflect the relative significance of the structure and may be a factor in planning decisions.
 Grade II accounts for 92% of listed buildings, while grade II* is an intermediate grade accounting for 5.5%; grade I holds the remaining 2.5% of listed buildings and is reserved for structures of exceptional significance.
 Grade I listed war memorials are deliberately very few, though several have been upgraded to grade I status as part of commemorations around the First World War centenary.
 A war memorial listed at grade I will be of exceptional interest for its design and artistic merit and will be of great historical interest.
 Such memorials are often the work of famous architects or sculptors, amongst the most prolific of whom was Sir Edwin Lutyens, whose memorials account for a third of all those listed at grade I. Lutyens designed dozens of war memorials across the United Kingdom and elsewhere in the Commonwealth, including the Cenotaph on Whitehall in London—the focus for the national Remembrance Sunday services—and the Arch of Remembrance in Leicester—the largest of Lutyens' war memorials in Britain; both are listed at grade I.
 As part of the commemorations of the centenary of the First World War, Historic England—the government body responsible for listing in England—is running a project with the aim of significantly increasing the number of war memorials on the National Heritage List for England.