Preheat the oven to 425°F (218°C).
In a medium bowl, whisk together the eggs, milk, salt, and black pepper until well combined.
Melt butter in a non-stick skillet over medium-low heat. Add egg mixture and cook, stirring frequently, until the eggs are scrambled and just set but still moist. Remove from heat immediately.
Place the pre-baked pizza crust on a parchment-lined baking sheet or a pizza stone.
Distribute the scrambled eggs evenly across the surface of the crust.
Top the eggs with the crumbled bacon and then the shredded cheddar cheese.
Bake for 8 to 10 minutes, or until the cheese is fully melted and the crust edges are golden brown.
Garnish with sliced green onions, slice into eight wedges, and serve warm.