On the second one I am planning to use a little bit more oil in the pan and a mix of white cheddar and whole milk mozzarella topped with pepperoni.
Ok, a few suggestions:
1. the oil amount you used seemed sufficient - too much and it gets weird
2. try thinly slicing about 1/3 of the cheese, covering the dough with it, then the sauce- the bottom cheese will keep the sauce from seeping into the dough and making it soggy
3. Dispense the blended sauce with a ketchup squeezer- this ducks the "smearing it around with a spoon" step, which can lead to isolated blobs of lots of sauce interspersed with heaped up mounds of dough
4. Use a cake knife to pry up and check the bottom browning. If the top is getting too done before the bottom is, open the oven. The residual heat in the stone will finish the bottom as long as the pan is sitting on it.