B737 pilot reported on departure experiencing a bird strike resulting in returning to the departure airport and being towed back to the gate.

Narrative

First leg of a 2-leg day; 5 day sequence. Takeoff normal; and on climbout at 1200' AGL I clearly saw a turkey vulture out the left window. A second later we heard what we thought we recognized as a bird strike (a hard knock) on the fuselage. I instinctively looked at the engine instruments and noticed no change from normal. About 5-10 seconds later the First Officer (FO) and I smelled a 'burnt hair' smell and seconds later were notified by FA's they smelled 'something burning' both of which indicated to us that the bird was ingested down the engine. We advised ATC of the bird strike; our intent to return to ZZZ; cleaned/leveled the aircraft at 3000'; and accepted vectors for ILS Runway XX. [Priority handling was requested]; Airport Rescue and Firefighting (ARFF) requested; flight attendants (FA's); company (ACARS); and passengers notified. We ran the Overweight Landing checklist; briefed threats and the approach with the intent to clear the runway if able. Per the Overweight Landing checklist we configured flaps 30 with autobrakes MAX. Landing was normal; cleared the runway and stopped the aircraft on Taxiway 1. We wer met by ARFF and we asked them to look over the aircraft; #1 engine; and brakes. Based upon the 175K pound landing and concern for overheating brakes we elected to coordinate for a tug into the gate. We shut down engines; tugged into the gate; then made aware of damage to the engine. Of note; there was no indication of engine damage or irregularity during flight which is why; although we suspected an ingested bird; I decided to keep the #1 engine running. The engine indicated and operated normally all of the way until shutdown.