Preheat oven to 500°; place a pizza stone on a rack in the lowest level of the oven.
Make the sauce: melt butter in a saucepan over medium heat; add in the garlic; cook 30 seconds or until lightly golden.
Add in the vodka; cook about 1 minute more until the alcohol evaporates.
Add in the tomatoes, crushed red pepper, and salt to taste; simmer 10 minutes.
Add in the cream and peas; cook until the sauce is reduced and thickened, about 5 minutes; let cool.
Place 1 ball of dough on a floured surface; holding you hands flat, pat the ball out evenly with your fingers, lifting it and turning it over several times, until it reaches a 12-inch circle (don’t knead the dough or handle it any more than is necessary; if the dough seems sticky, dust it lightly with flour.
Dust a pizza peel or rimless cookie sheet with flour.
Carefully transfer the circle of dough to the peel, reshaping it as needed.
Shake the peel once or twice to make sure the dough does not stick.
If it does, sprinkle the peel with more flour; quickly top the dough with the sauce, spreading it to within ½ inch of the edge with the back of a spoon.
Layer with half the mozzarella, mushrooms, and a sprinkling of pecorino.
To slide the pizza onto the baking stone, line up the edge of the pizza peel with the back edge of the stone, then tilt the peel, jerking it gently to start the pizza moving.
Once the edge of the pizza touches the stone, carefully pull back on the peel until the pizza is completely off.
After the pizza is on the stone, do not attempt to move it until it firms up in 2-3 minutes.
Bake the pizza 6-7 minutes, or until the cheese is melted and the crust is browned; just before serving, sprinkle with the prosciutto and basil.