Date: 202110
Local time of day: 0601-1200
Location: ZZZ.TRACON, US
Flight conditions: VMC
Light: Daylight
Aircraft: B737-800
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class B ZZZ
ATC / advisory: TRACON ZZZ
Aircraft component: Turbine Engine
Component problem: Malfunctioning
Reporter organization: Air Carrier
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Troubleshooting
Second reporter function: First Officer; Pilot Flying
Anomaly: Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Automation Aircraft Other Automation; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Flight Crew Inflight Shutdown; Flight Crew Overcame Equipment Problem; Flight Crew Returned To Departure Airport; General Flight Cancelled / Delayed; General Maintenance Action
Contributing factors: Aircraft; Human Factors
Primary problem: Aircraft

Narrative: While climbing through 17;000 feet on the ZZZZZ1 Departure from ZZZ; I saw that the #1 engine N1 was indicating 106% and was highlighted red. I pointed out the N1 overspeed to my First Officer who was the Pilot Flying. I informed ATC that we needed to level off due to an engine problem. I got out the QRC and we ran the Engine Limit or Surge or Stall Checklist. After the thrust lever was brought back to idle (step 2); the N1 remained in the overspeed condition. We transitioned from the QRC to the QRH and continued with the checklist. With the thrust lever at idle for perhaps 20 seconds; as I was pulling out the QRH; the N1 very slowly started decreasing. However; after decreasing below 100%; it began to increase again. At this point; per the QRH; we shut the engine down. We requested priority handling and returned to ZZZ. While the First Officer was handling ATC communications and flying; I briefed the Flight Attendants of the situation.I told them that we had to shut an engine down; that we would return to ZZZ in 20 minutes; and they did not need to prepare the cabin. I briefed the Passengers on what had happened and that we were returning to ZZZ. I was very cognizant of the Passengers anxiety and did my best to assuage their angst. I called ZZZ Ops on 131.7 and asked them to get Dispatch on the line. After several minutes ZZZ Ops informed me that they were unable to get Dispatch. We retrieved landing and brake cooling data. After completing the Engine Limit or Surge or Stall Checklist; we completed the One Engine Inoperative Landing Checklist. I took control of the aircraft about 12 miles out and we landed uneventfully on 36L. We rolled out to the end and taxied onto Taxiway 1. The ARFF vehicles met us; chocked our nose tires and safely measured our brake temperature every seven or eight minutes. After 30 minutes (required brake cooling time) we taxied back to gate.


Second narrative: Climbing through 17;000 feet; the #1 Engine N1 surged to 106%. Captain completed QRC and required actions for 'Engine Limit or Surge'.  Briefly; the N1 appeared to lower (slowly to 97%) only to rise again as the Captain began to read through the QRH.  I was the Pilot Flying and handled ATC communications for vectors and a request to return to ZZZ.  Captain completed all required checklists; weight and balance calculations for landing speeds and brake cooling; communicated with flight attendants; passengers and Dispatch.After completing all checklists; controls were transferred to the Captain approximately 12 miles out on an extended right base leg to Runway XXL in ZZZ.  Aircraft landed safely in ZZZ with gentle touchdown rate due to overweight landing considerations.  After required brake cooling time of 30 minutes was complied with; we taxied to the gate and Passengers were re-accommodated.  Afterwards; the Captain and I spoke with Dispatch; Maintenance Control; Scheduling; and on-site (Company) Maintenance Personnel. After conferring with Chief Pilot in the Operations Coordination Center; it was agreed that our day was complete and we should make our way back to ZZZ1.


Synopsis: B737-800 flight crew reported an uncontrollable engine N1 speed; causing an in flight shut down of the affected engine. Flight crew conducted an air turn back to a precautionary landing.