Preheat your oven to 425°F (220°C).
In a frying pan over medium heat, add the breakfast sausage. Cook, breaking it apart with a spatula until browned and cooked through, about 5-7 minutes. Drain any excess fat and set aside.
In a mixing bowl, whisk together the eggs, milk, salt, and black pepper until well combined.
In the same frying pan, heat the olive oil over medium heat, then pour in the egg mixture. Cook gently, stirring occasionally, until the eggs are scrambled but still slightly runny, about 4-5 minutes. Remove from heat.
Roll out the pizza dough on a floured surface to your desired thickness, then transfer it onto a baking sheet or pizza stone.
Spread the cooked sausage evenly over the pizza dough, followed by the scrambled eggs, and then top with shredded cheese.
Bake the pizza in the preheated oven for about 10-12 minutes, or until the crust is golden brown and the cheese is melted.
Remove from the oven, let it cool for a minute, then slice and garnish with chopped green onions or parsley if desired.