B737-800 Captain reported engine problems caused an air turn back and precautionary landing during a post heavy maintenance functional check flight.

Narrative

During the FCF (Functional Check Flight) engine relight profile with #1 engine shutdown; the alternate gear and LGTU (Landing Gear Transfer Unit) checks were successfully completed on descent phase of the profile and #1 engine windmill relight was attempted passing 23;500 feet and 305 KIAS approximately 45 NM Northeast of ZZZ. The blue Engine Valve light on the fuel panel remained illuminated when the #1 engine start lever was placed to run with no Fuel Flow. EGT (Exhaust Gas Temperature) rise or indication of relight. The FCF profile was terminated. QRH (Quick Reference Handbook) procedures (Engine In-Flight Start via Cross bleed) [were] accomplished followed with requesting priority handling as we turned back towards ZZZ. The #1 engine did restart with a cross bleed attempt and QRH procedures [were] completed followed with the QRH precautionary One Engine Inoperative Landing checklist & Non-routine; landing assessment calculation. The descent checklist and pre-landing checklist completed on descent. An uneventful flaps 15 approach and landing conducted to Runway 18L followed with a normal taxi to the Company Maintenance and Engineering ramp. Debriefed with Maintenance. For the [flight] I conducted a few days later; maintenance advised the HMU (Hydro Mechanical Unit); EEC (Electronic Engine Control) alternator and EEC was replaced and normal FCF windmill restart of #1 was successfully accomplished. The rest of the FCF profile for the aircraft Main Base Visit Heavy Check was completed.Faulty engine component; either HMU (Hydro Mechanical Unit); EEC and EEC alternator as all three replaced at once. No added suggestions as the pilot group trains to a wide range of contingencies; mitigates risk and prepared to terminate the profile while complying with QRH and normal procedures.