B737 Captain reported loss of thrust and engine vibration after selecting Engine Anti Ice to on. The Flight Crew performed an inflight shut down of the affected engine and made an air turn back and precautionary landing at destination airport.

Narrative

Departed Runway XXL at ZZZ with radar vectors to the south. FO (First Officer) was PF (Pilot Flying); I was PM (Pilot Monitoring). Encountered moderate turbulence climbing through approximately FL 190. I called FA (Flight Attendants) to have them remain seated for a while longer. I observed that we now in icing conditions (had entered cloud with OAT of 5 Degrees C.); so I turned on Engine Anti Ice. Approximately 15 seconds later while climbing through FL 220; #1 Engine had substantial loss of thrust with loud noise and vibration. Aircraft yawed sharply left. FO was still hand flying at this point. He immediately announced 'My Aircraft'; applied right rudder & aileron while reducing aircraft pitch attitude to maintain airspeed. I instructed him to level off and regain control of aircraft. Notified ATC that we wanted to stop the climb and descend back to FL 210. Then turned attention back to #1 Engine. Was producing about 35-40% N1 while still making noise and vibrating. FO disconnected Auto throttles and retarded #1 thrust lever to idle. Noise and vibration stopped with engine still running at flight idle. #1 FA called to ask if we had heard the noise. I told him that we had an engine problem; that the aircraft was under control; and that I would get back to them as soon as I could. I [requested priority handling] with ATC and requested vectors back to ZZZ. ATC began vectors with an initial right hand turn to the southwest; followed by a turn to the west. I told controller we would need a decent sized box pattern back to ZZZ to accomplish checklists. Received ATC clearance back to ZZZ via radar vectors. FO suggested descent to exit icing conditions. I made the request with ATC and we descended back into clear air at 17;000 feet; where FO trimmed the aircraft for single-engine cruise. I suggested an attempt to see if the engine would now produce any more power. As the FO began to advance the #1 thrust lever; the noise and vibration returned without any increase in thrust. After discussing the situation with the FO; I made the decision to secure the engine. I used the QRH to accomplish engine severe damage or separation; one engine inoperative landing; and non-routine landing Checklists. This included briefing the flight attendants for a precautionary landing; making 2 PAs to the passengers; and notifying Dispatch via the Crew Phone. Received radar vectors for ILS to Runway XX in VMC. Confirmed with ATC that ARFF (Airport Rescue and Firefighting) would be standing by.FO was still PF while I also made all necessary modifications to the FMC; including route change and setting up for the approach. Completed normal Descent checklist and Before Landing checklist to the line. Once established on final approach; I assumed control of the aircraft and became the PF to accomplish the landing. I made this decision because I had much more experience in the 737 (new CA with just 130 hours but flew it for 2 years an FO with about 1400 hours). The new hire FO had just recently completed his OE and had just over 40 hours in the 737. We had identified the low-time status for each of us as a threat in our earlier briefing at the gate. Configured for single-engine landing and completed Before Landing checklist. Performed single-engine landing on Runway XX at ZZZ. Stopped aircraft on the runway at approximately the mid-field position (near taxiway F). Made a quick PA instructing passengers to remain seated as Safety vehicles completed an inspection. ATC assigned discreet frequency for us to communicate with ARFF Commander. Had ARFF conduct visual inspection of the aircraft. Received the 'all clear' and then taxied to gate with ARFF following us. Upon arriving at gate; made a final PA to passengers advising them that the aircraft would be going out of service and to take all belongings as they exited; and that a gate agent would have more information concerning the flight as a new plan was put together. Gate agent came on board and informed me thatthey had a new aircraft and gate. I informed passengers of the new gate with a final PA. Made an AML (Aircraft Maintenance Logbook) entry for the engine failure and discussed with Maintenance Supervisor; who had also come on board. I want to commend the performance of the New Hire FO. He did an outstanding job of maintaining aircraft control and communicating with ATC when I was otherwise busy talking to other involved parties. While new to airline with just over 40 hours in the 737; his previous experience as a different make/model pilot proved invaluable. I think we worked together very well by implementing the company strategy and communicating effectively. It proves that 2 pilots who were flying just our 2nd leg together can 'gel' and successfully work through a serious situation.