Why does my mustang shut off after 15 min?

hook up a fuel pressure guage. let it run and when it quits see if its fuel pressure that you are loosing. if not then you must be losing spark.

Nice car and good, reliable powertrain. kbro is most likely correct since electrical components (like fuel pumps) heat up, lose contact, cool off and then work again. Any other concerns? Check engine light, ping or backfire (muffled pop sound) on acceleration? You can try listening for the fuel pump when it's cooled off and when it fails - it doesn't always indicate a bad pump but it can. When the car has been sitting a while, turn off all the accessories and roll the windows up. Turn the Ignition to "ON" and listen for a humming in the back of the car - it only lasts for 3 seconds when the engine is off so you may have to do it several times. That hum is the fuel pump. Run the vehicle until it stops again and check for the sound again the same way. No noise - no fuel.