Preheat the oven to 350°F (175°C).
In a large skillet, sauté the diced onion and bell peppers in a little olive oil until softened, about 5 minutes.
Add the diced potatoes to the skillet and continue to cook until they are slightly tender, about 10 minutes, stirring occasionally.
In a large bowl, whisk together the eggs, milk, salt, and black pepper until well combined.
Stir the sautéed mixture into the egg mixture, followed by the diced ham and half of the cheddar cheese. Combine thoroughly.
Pour the combined mixture into a greased baking dish and spread it out evenly. Top with the remaining cheddar cheese.
Bake in the preheated oven for 30–35 minutes or until the cheese is bubbly and the center is set.
Remove from the oven and let stand for 5 minutes before slicing and serving.