Car shuts off will not start until waiting ten minutes or so

So, I have had this car for almost a year and have replaced pretty much everything imaginable with it. This issue I have been having for a few months has prevented me from driving the car. I live in Arizona, so temperatures reach 118 outside often. The issue is; I will be driving the car and after about 15 minutes of driving it or so it will straight up shut off. It will just stop going. Of course then the oil and battery lights go on (which I know is simply because the car is off as it has shut off). If I try and turn it back on, there is no sound, it just won't turn on. The lights, radio, A/C, everything else continues to work but the car just won't go. If I wait about 10 minutes with the car off it will start again just fine. My mechanic originally thought it was the radiator which we replaced, and then the ignition module which we also replaced, but to no avail the issue still persists. I love this car and do not want to sell it. Please help!