Preheat the oven to 350°F (175°C). Grease a 9-by-13-inch baking dish.
Cook the onion and bell peppers in a lightly oiled skillet for 3 to 4 minutes. If using raw sausage, brown it fully and drain before adding it to the casserole.
Whisk the eggs, milk, salt, pepper, and optional paprika in a large bowl.
Stir in the cooked sausage, vegetables, hash browns, and greens. Fold in the cheddar, mozzarella, and Parmesan, reserving a little cheese for the top.
Pour into the prepared dish and sprinkle with the reserved cheese.
Bake for 40 to 45 minutes, until the center is set and a knife inserted near the middle comes out clean.
Rest for 5 to 10 minutes, then top with sliced green onions and serve.