I don't think a pizza stone has the thermal mass needed to cook a pie in the way you describe. It probably is getting very hot, but when you slap the cold dough on top of the stone there isn't enough heat stored to properly cook the bottom of the pizza. The floors of wood burning ovens are usually several inches thick, which allows them to absorb massive amounts of heat energy so they don't cool down so much when a raw pizza is placed on them. With something as thin as a pizza stone, you would need some kind of heat source underneath to bring the temperature back up enough to cook the bottom of the dough. I wonder if you could set up another stone over a gas burner on your stove top to cook the bottom of the pizza after the top has cooked under the broiler.