First thing to do is adjust the internal pushrod between the booster and the master cylinder. Shorten the rod by only one turn and try it. It is a very fine thread and one turn makes much difference in the engagement distance.
If one turn is not enough, then do it again with one more turn.
Once the brake components under the hood cool down, the problem goes away for a while, then as they heat during driving, the problem comes back...right?