Heat oven to 425° F with a rack in the lowest position. Coat a 9-by-13-inch baking dish with 1 tablespoon of the oil.
Press the dough into the pan, covering the bottom and sides. Top the dough with the pesto, cheese, sausage, and Brussels sprouts. Drizzle with the remaining 1 tablespoon of oil.
Bake until the crust is golden brown and the sausage is cooked through, 30 to 35 minutes. Let the pizza rest for 5 minutes before serving.