Air carrier Captain reported on departure one of the engines indicated a high Exhaust Gas Temperature. The flight crew retarded power to idle on the engine and returned to the airport for an overweight landing.

Narrative

ATIS reported Low Level Windshear; therefore; we selected Windshear Profile. After takeoff; at approximately 300 ft. to 400 ft.; I noticed the #1 Engine EGT went red and indicated 968 degrees. I incorrectly stated we had an engine overheat because I saw the high EGT. At 1000 ft. after cleaning up the aircraft; I asked the F/O (First Officer) to pull out the QRH for Engine Overheat and as he looked at the index; he realized that it was a QRC item (as was the checklist we should have run: Engine Limit or Surge or Stall). We ran the Engine Overheat QRC and QRH. Everything with the wrong QRC/QRH went as I was expecting with the one exception of it having us retard the thrust lever all the way to idle rather than to the point where the indication went away as I was expecting; however; I clarified with the F/O that that was what it said; and we again failed to trap the error. The red indication went away shortly after reducing the thrust lever. We informed Tower we would be returning to ZZZ; due to a malfunction with our #1 engine. I notified the flight attendants of the situation and that we did not anticipate an evacuation. I sent an ACARS message to Dispatch to let them know of the situation and that we were returning to ZZZ. I also requested that they get us a gate. We had requested with ATC slightly extended vectors to Runway XX; in order to accomplish all the Non-normal and Normal Checklists and to set up and brief the approach and landing. We [requested priority handling] with ATC; so that the Crash Fire Rescue would be on hand; in case we had any further engine trouble and due to the fact that we would be landing overweight in a high ambient temperature of 99 degrees F. We decided to perform a 15 Flap landing in case we had to go around and had any further trouble with the #1 Engine; especially due to the high ambient temperature with a fully loaded aircraft. We also elected to use autobrakes 3. The landing was smooth; and we rolled out to Taxiway 1 at Tower's request; so that we could pull off into the de-ice pad for the fire trucks to inspect the aircraft. We ran the brake cooling performance data. We had to allow 30 minutes of cooling before returning to the gate. We coordinated with Crash Fire Rescue to shut down the engines so that they could chock the nose wheel for us to release the parking brake and help with brake cooling. At this point; we debriefed how we handled anything and as we reviewed the QRH again on the ground the F/O noticed that the indications section of the QRH showed the Engine Overheat lights and that that did not match our situation. This is when we realized we should have run the Engine Limit or Surge or Stall QRC and QRH. After the 30 minutes brake cooling; we coordinated to have the chocks removed and restarted the #2 Engine. Crash Fire Rescue followed us taxiing. We ensured Ground Personnel would be aware that #2 Engine would be running instead of the usual #1 Engine until we were stopped at the gate.After selecting the wrong QRC and continuing into the QRH; we should have re-verified the indications shown above the QRH steps. This would have shown us that we were in the wrong checklist. Additionally; I should have queried more than I did when I felt that retarding the thrust lever to idle rather than just until the indication went away. I should have had us re-verify the indications/condition section of the QRH. Lastly; I think consideration should be given to removing these two emergency checklists from the QRC and to only have them in the QRH. This would drive the first actions we take to be verifying the indications prior to following an initial wrong selection that gets us headed down the wrong path. Alternatively; I think adding the indications to the QRC; would help limit this kind of error.