Whisk eggs and milk together in a bowl; season with salt and pepper.
Spray a large skillet with cooking spray; heat over medium-high heat. Pour egg mixture into the skillet; cook and stir until eggs are set, about 5 minutes. Add sausage, bacon, tomatoes, and mushrooms, and cook until vegetables are soft, about 3 minutes.
Spread scrambled mixture over flatbread; top with shredded Monterey Jack cheese.
Bake flatbread pizza in the preheated oven until the cheese is bubbly and underside of the crust is golden brown, 5 to 7 minutes.