B737 Captain reported an engine vibration and EGT exceedance during climb. The crew returned to the departure airport and landed safely.

Narrative

Taxi and takeoff were uneventful; although on taxi-out; ZZZ Tower advised us of LLWS +/- 20 knots on final for [Runway] XXL. the FO (First Officer) was PF (Pilot Flying); and I was PM. The FO asked how ATC had that information in the absence of a PIREP. ATC explained that they used technology. We departed XXL with an assigned heading of 150 and a clearance to climb to 3000 ft. On a right downwind; were switched to departure. At that moment; just south of ZZZ1; the right engine experienced severe vibration accompanied with abnormal N1 and EGT indications. The EGT indicated an exceedance and displayed a red 'disk'. ATC cleared us to 6000 ft; but I set the altitude in the MCP (Mode Control Panel) to 3000 ft and advised Captain to maintain heading. It took several seconds to transmit to ATC; due to multiple aircraft in the approach/departure corridor. I advised ATC and requested immediate return to ZZZ. ATC asked if we were ready to return right away; and I informed them that we needed vectors to a safe area to allow us to run a checklist; then we would advise them. I started the Engine Vibration Checklist; then stopped. I reminded Captain that we had an exceedance; and the correct checklist was the Engine Fire; Failure; Severe Damage; or Separation QRC; followed by the QRH. We followed both the QRC and QRH deliberately and methodically; and once we were on the Deferred Items of the Engine Inoperative Landing Checklist; I requested vectors to the ILS XY Approach; and briefed the visual approach backed up by the ILS. We were landing Flaps 15; using speed off the INIT REF page; after verifying that we were not overweight. ATC advised us of LLWS on final approach. I referred to PROG PAGE 2 to ensure that we were not landing with a tailwind component. It was difficult to slow the aircraft down to VREF at Flaps 15; so I extended the speedbrake 30% to create drag. (The limitation for use of speedbrakes are up to Flaps 10). I explained to Captain that I had to 'hang some tin' to get the speed under control; and we did not want excess kinetic energy. We landed with Autobrakes 3; taxied off at taxiway 1; and were instructed to wait at the holding pad. The ARFF (Airport Rescue and Firefighting) inspected the exterior of the aircraft; especially the right engine; then took the brake temperature (700C). After 30 minutes; we were cleared to be towed to the ramp. The Flight Attendants performed an excellent job of controlling the Passengers and keeping them calm. There were only XX people onboard; but not everyone spoke English; so that was a challenge. I kept the Passengers and Crew informed during the whole adventure; explaining that we would return to ZZZ and exchange this aircraft for a 'newer one'. I did this in English and in Spanish. My F/O performed his PF tasks flawlessly; displaying superb airmanship and reflecting the quality of his training. The Flight Attendants are to be commended on their professionalism and actions; which kept all of our customers safe from the moment of the fire to the deplaning at the gate.Good preparation and constant drills in the simulator; along with standardized briefings and adherence to SOPs made this event resemble a training exercise rather than a full-blown emergency. Every Crew Member did their jobs flawlessly. My only exception was the deliberate use of speedbrakes at Flaps 15; when the SOP prohibits their use above flaps 10. I chose this course of action because the risk of damaging the flap/speedbrakes was outweighed by the risk of a high-energy touchdown and a possible overrun; or the risk of a single-engine go-around in conditions of low-level windshear.