Preheat oven to 500 degrees F, heating pizza stone, if using.
Dust a wooden cutting board with flour, place dough on flour and stretch to desired size. Toss sprouts and onions in 1 tablespoon olive oil. Drizzle dough with 1 teaspoon rosemary-infused olive oil and top with sprouts and onions. Transfer pizza to pizza stone or baking sheet and bake until dough sets, 10 minutes.
Carefully crack eggs into a liquid measuring cup. Remove pizza from oven and pour eggs into the center; return pizza to oven and continue baking until the crust is brown and eggs are set, rotating as needed, two minutes.
Remove pizza from the oven, drizzle with remaining 2 teaspoon rosemary-infused olive oil, Parmesan, oregano, freshly ground black pepper, and a pinch of Maldon salt on the yolks.