Adjust rack to upper-middle position and heat oven to 500 degrees. Grease a 12-inch oven safe skillet with 1 tablespoon oil.
Pulse drained tomatoes, garlic, salt, red pepper flakes, and 1 tablespoon olive oil together in a small food processor until coarsely ground. Reserve.
Place dough on lightly floured counter. Press and roll dough into an 11-inch round, so it can fit inside a 12-inch pan nicely. Transfer dough to prepared skillet; reshape as needed. Spread as much sauce as you want over dough, leaving ½-inch border at edge. Sprinkle most of the mozzarella evenly over sauce. Add shallots and mushrooms evenly. Crumble up the sausage out of the casing and spread it evenly on top, with your fingers. Top with remaining mozzarella and Parmesan.
Set skillet over high heat and cook until outside edge of dough is set, pizza is lightly puffed, and bottom crust is starting to get cooked when gently lifted with spatula, about 3-4 minutes.
Transfer pizza to oven and bake until crust is brown, bottom is crisp, sausage is cooked, and cheese is nicely melted, 7 to 10 minutes. Carefully remove skillet from oven (pan and handle will be very hot). With a flat spatula, carefully lift pizza out of the pan and onto a cutting board. Sprinkle with fresh sliced scallions for garnish. Let pizza cool for a couple minutes before slicing and serving.