Preheat the oven to 375°F (190°C). Fit the pie crust into a 9-inch pie dish and prick its base lightly with a fork.
Warm the olive oil in a skillet over medium heat. Cook the onion and bell peppers for 4 to 5 minutes, until softened. Let cool slightly.
Whisk the eggs, milk or half-and-half, salt, and black pepper until smooth.
Scatter the ham, sautéed vegetables, and cheddar evenly over the crust.
Pour the egg mixture over the fillings, distributing them gently with a spatula.
Bake for 35 to 45 minutes, until the top is golden and the center is set.
Rest for 10 to 15 minutes before slicing and serving.