Preheat oven to 475°F. Place a pizza stone or upside-down sheet pan inside to heat.
Cook sausage in a skillet over medium heat until browned, 5–6 minutes. Stir in bell pepper for 2 minutes. Drain excess fat.
Whisk eggs with milk, half the salt, and black pepper.
Soft-scramble eggs in a lightly oiled skillet over low heat until just set, 2–3 minutes. Remove from heat.
Stretch dough on parchment into a 12–14 inch round. Brush with olive oil and sprinkle garlic powder and remaining salt.
Scatter half the mozzarella on the dough. Add sausage-pepper mixture, then spoon on the soft eggs. Top with remaining mozzarella and cheddar. Add red pepper flakes if using.
Slide parchment onto the hot stone or pan. Bake until the crust is deep golden and cheese bubbles, 12–14 minutes.
Finish with green onions and parsley. Slice and serve hot.