B737 Captain reported an engine temperature exceedance during climb out and a return to departure airport; requiring an overweight landing.

Narrative

On climb out; while the after takeoff checklist was being performed; I noticed an EGT exceedance on the number one engine. I transferred aircraft control and radios to the FO (First Officer). We performed the memory items and QRH; which directed us to keep the engine at a reduced power setting in order to keep the EGT within limits. I attempted to contact Dispatch and Maintenance Control; but the ARINC frequency was not available- upon tuning the frequency I heard another aircraft beginning a serious med link case. ARINC advised that no other frequency was available. The FO and I discussed the situation. Considering we could not operate the number 1 engine at climb power without an exceedance; and a discussion with Maintenance Control and Dispatch was impractical; we agreed the most prudent decision would be to return to ZZZ. I explained the situation to Dispatch via ACARS; notifying the Dispatcher that we would be returning to ZZZ; and began the diversion process. I gave the flight attendants a briefing; stating that there was no need to prepare the cabin for an evacuation. I made a PA to the passengers; notified ATC and were cleared back to ZZZ. Initially we were not going to request priority handling; but because we would be landing overweight and were requesting the longest Runway; XXR; priority handling was requested. We coordinated fire rescue to meet the aircraft. Dispatch advised that we were near our performance limit for bleeds on; so we configured for a bleeds off landing per the FM (Flight Manual). We transferred control again and I took control of the aircraft. We landed normally. After landing; we cleared the runway and fire rescue inspected the brake temps. We taxied to the gate without incident; and I made an logbook entry for the overweight landing and EGT exceedance.