Preheat the oven to 500°F with a rack set in the bottom position.
Grease a large (13 x 18-inch) rimmed baking sheet with the olive oil. Place the dough in the pan and press it out with your fingers until it reaches all the way to the edges and corners of the pan. If the dough keeps shrinking back, let it rest for 10 to 15 minutes and try again. After the dough is pressed out all the way to the edges of the pan, cover the whole pan with plastic wrap or a clean kitchen towel and let it rest (proof) for 30 minutes.
Cook the sausage in a medium skillet over medium heat until it is no longer pink, 5 to 7 minutes. As the sausage cooks, use a wooden spoon to break it up into small, bite-size pieces. Remove the skillet from the heat.
When the dough has finished proofing, scatter all but ½ cup of the mozzarella evenly on top. Use a large spoon to dollop the sauce over the cheese, and gently spread it out in an even layer. Top with the sausage, pepper, onion, and mushrooms, season with a couple pinches of salt and a few grinds of black pepper, and scatter on the remaining ½ cup of mozzarella and the finely shredded aged gouda cheese.
Transfer the baking sheet to the oven. Bake the pizza for 15 minutes, rotating the pan once halfway through, until the bottom of the crust is golden brown. (Use a spatula to carefully lift up one corner of the pizza to check underneath.)
Remove the pizza from the oven. Let it cool for 5 minutes, then cut into squares and serve.