Inclusion Criteria:  Willingness to comply with study procedures/requirements Capable of giving informed consent Diagnosis of at least one PG ulcer by clinical, histological and laboratory assessments with a minimum wound size of 4 cm2. Male age 18-99 who agree to not father a child or donate sperm while on study and at least 1 week following last dose of the study drug. If subject is a sexually active male and could cause a pregnancy, subject must be sure that female partner(s) are using birth control that works well or not have sex. Female age 18-99; either of non-childbearing potential or of childbearing potential who test negative for pregnancy and agree to use at least two reliable methods of birth control or remain abstinent during the study and for at least 1 week following the last dose of baricitinib. Classic PG defined as deep ulceration with undermining violaceous borders. Are candidate for systemic therapy. All participants will be taking and clinically stable 30 mg (same ulcer size) of prednisone fort least two weeks at the start of the study. Patients must have discontinued immunosuppressive therapies (cyclosporine, azathioprine, mycophenolate mofetil, methotrexate, apremilast, dapsone) due to inadequate response or intolerance for at least 4 weeks prior to beginning the study drug. Undergoing at least once a week standard of care wound care at home or wound care facility. Willingness to travel to Oregon Health and Science University (OHSU) for all study visits, or living >30 miles from OHSU and willing/able to participate in remote videoconferencing visits with access to a computer with internet and webcam capabilities.  Exclusion Criteria:  Have history of malignancy or lymphoproliferative disease; or have signs or symptoms suggestive of possible lymphoproliferative disease, including lymphadenopathy or splenomegaly; or have active primary or recurrent malignant disease; or have been in remission from clinically significant malignancy for < 5years. Patients with cervical carcinoma in situ, basal cell carcinoma or squamous cell carcinomas who have had active disease within 3 years of screening for this study. Active, untreated, acute or chronic infection (such as untreated tuberculosis), or immunocompromised to an extent that such that participation in the study would pose an unacceptable risk to the subject. (Treated infections such as latent tuberculosis after completion of the appropriate therapy are not excluded.) Clinically serious infection or received intravenous antibiotics for an infection, within 4 weeks of randomization. Active viral infection that, based on the investigator's clinical assessment, makes the subject and unsuitable candidate for the study. Positive for human immunodeficiency virus, hepatitis B virus, or hepatitis C virus. Symptomatic herpes zoster infection within 12 weeks of screening or recurrent or disseminated (even a single episode) herpes zoster. Symptomatic herpes simplex at the time of randomization or disseminated (even a single episode) herpes simplex History of disseminated opportunistic infections (e.g., listeriosis and histoplasmosis).  Have a history of venous thromboembolism (VTE), or are considered at high risk for VTE as deemed by the investigator, or have 2 or more of the following risk factors for VTE:  Aged >65 years. Body mass index (BMI) >35 kg/m2. Oral contraceptive use and current smoker. Estimated glomerular filtration rate (eGFR) <60 mL/min/1.73 m2 (Chronic Kidney Disease Epidemiology Collaboration equation Creatinine 2009 equation). Wound care debridement of any PG ulcer within 2 weeks. Intralesional corticosteroids within 4 weeks of screening. Previous exposure to a Janus kinase (JAK) inhibitor (ruxolitinib, tofacitinib, upadacitinib, filgotinib). For biologic therapies, the specific washout periods used will at least 4 weeks for anakinra, etanercept, infliximab, adalimumab, alefacept, golimumab, secukinumab, ixekizumab, risankizumab, guselkumab, tildrakizumab, canakinumab, ustekinumab and at least 24 weeks for rituximab or efalizumab. Patients who are currently on immunosuppressive therapies or have not discontinued them for at least 4 weeks. Clinically significant (per investigator's judgement) drug or alcohol abuse within the last 6 months preceding the Baseline Visit. Have a live vaccine within 12 weeks prior to baseline or intend to have a live vaccine during the course of study. Had any major surgery within 8 weeks prior to baseline or will require major surgery during the study, which in the opinion of the investigator would pose an unacceptable risk to the subject. Recent (within past 6 months) cerebrovascular accident, myocardial infarction, coronary stenting. Uncontrolled hypertension - confirmed systolic blood pressure >160 mmHg or diastolic blood pressure >100 mm Hg. Gastrointestinal (GI) perforation (other than appendicitis or penetrating injury), diverticulitis or significantly increased for GI perforation (within past 6 months) per investigator judgement; any condition could interfere with drug absorption including but not limited to short bowel syndrome. Presence of significant uncontrolled respiratory, hepatic, renal, endocrine, hematologic, neurologic, or neuropsychiatric disorders, or abnormal laboratory screening values that, in the opinion of the investigator, pose an unacceptable risk to the subject if participating in the study or of interfering with the interpretation of the data. Have clinical laboratory test results at screening that are outside the normal reference range of the population and are considered clinically significant, or have any of the following specific abnormalities: Neutrophil count <1500 cells/µL, lymphocyte count <500 cells/µL, platelet count <100,000 cells/µL, aspartate transaminase (AST) or alanine aminotransferase (ALT) > 2 times the upper limit of normal, hemoglobin <10 g/dL for male and female subjects, eGFR>60. Women who are lactating or breastfeeding. Have any other condition that precludes the subject from following and completing the protocol, in the opinion of the investigator. Are investigator site personnel directly affiliated with this study and/or their immediate families (spouse, parent, child, or sibling). Are currently enrolled in, or discontinued from a clinical trial involving an investigational product or non-approved use of a drug or device within the last 4 weeks or a period of at least 5 half-lives of the last administration of the drug, whichever is longer, or concurrently enrolled in any other type of medical research judged not to be scientifically or medically compatible with this study.