Prep the pan: Heat the oven to 350°F (175°C). Grease a 9x13-inch baking dish with nonstick spray or butter.
Sauté the veggies: Warm the olive oil or butter in a skillet over medium heat.
Add the onion and cook 3–4 minutes until softened. Stir in the diced peppers and cook another 4–5 minutes until tender. Remove from heat and let cool slightly.
Whisk the eggs: In a large bowl, whisk the eggs, milk, salt, black pepper, garlic powder, and smoked paprika until smooth and evenly combined.
Add the mix-ins: Stir in the sautéed peppers and onion, diced ham, and 1 1/2 cups of the shredded cheese.
Reserve the remaining 1/2 cup of cheese for topping.
Assemble: Pour the egg mixture into the prepared baking dish. Smooth the top and sprinkle with the remaining cheese.
Bake: Place on the center rack and bake 30–40 minutes, until the edges are set and the center is just barely jiggly. A knife inserted near the center should come out mostly clean.
Rest and garnish: Let the egg bake rest for 10 minutes before slicing.
Top with chopped parsley or chives.
Serve: Cut into squares and serve warm. Add hot sauce, avocado slices, or a light salad on the side if you like.