Air carrier Captain reported engine overheat indication on initial climb and return to the airport for an overweight landing.

Narrative

On preflight check of Logbook; the previous flight had shutdown number 1 engine for engine overheat on taxi in. Balancing entry from maintenance was the switch for the number one engine was in the wrong (placarded) position. Taxi and takeoff were normal; with the FO as pilot flying. During the climb out passing through approximately 10;000 feet; the number one engine overheat light illuminated. My first thought was we had the wrong position selected on the number on engine. After confirming the switch was in the correct position and confirming it with the FO; we ran the QRH procedure for engine overheat and the light extinguished when the thrust lever reached idle. We elected to return overweight to ZZZ for the ILS XXR. I took control of the aircraft on extended base we and stopped on the runway for ARFF (Airport Rescue and Firefighting) to check for heat signature on the number one engine and the brakes. The landing was normal. ARFF reported the brakes were 800 degrees.We taxied clear of the runway and looked at the brake cooling chart; talked with ZZZ maintenance and Maintenance Control and had ARFF take another temperature reading. The brakes had cooled significantly and we were escorted to the gate by ARFF.The main issue was not the engine overheat light; but the overweight landing and how to handle the hot brake issue. ARFF gave us a numerical temperature; we have no brake temperature reading on the flight deck nor any reference to actual temperatures for hot brakes. The chart for brake cooling is for rejected takeoff. Maintenance had no information in their manuals to offer guidance on when it was safe to taxi to the gate. I was concerned about blowing a tire(tires) in the gate area. As we began to taxi; the brakes felt and sounded normal and ARFF was convinced the best course of action was to get to the gate and get fans on them. It would be beneficial to have some notes for guidance in the overweight landing QRH procedure; maybe a temperature range that we can use the information that ARFF gives us.