B767-300 and 300 ER
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Aircraft Equipment Problem Critical; Inflight Event / Encounter Fuel Issue
Air Traffic Control Provided Assistance; Flight Crew Overcame Equipment Problem; Flight Crew Requested ATC Assistance / Clarification
Aircraft
Preflight; takeoff and initial phases of flight were normal and uneventful.  While at FL380 and approximately 150 miles west of ZZZ; I decided to empty the center tank of 800 PDS that had migrated during the flight. Also we had a 900 PD imbalance between the left and right wing tanks that I wanted to equalize.  Just seconds into the cross feed we got a 'Left fuel system Lo Press' EICAS immediately followed by the left engine rolling back and flaming out.  I took control of the aircraft and First Officer  retrieved and ran the In-flight Engine Restart Checklist.  Simultaneously; I coordinated with ATC a decent to FL340.  By the time we reached [flight level] 340 the engine was restarted and at idle with all indications being normal we slowly brought the power to a cruise setting.  All indications remained normal and within parameters.  Aircraft control was transferred back to First Officer and I contacted [Operations Control]; [Maintenance Control] and Captain (On Duty) via Airinc. After conferencing with all; we agreed the best and safe course was to continue to ZZZ1.  The flight continued and terminated in ZZZ1 without further incident.
B-767-300 pilot reported inflight engine shutdown during cruise.