Preheat oven to 350 degrees F (175 degrees C). Line a 9-inch deep-dish pie pan with crust.
Melt 2 tablespoons butter in a large saucepan over medium-low heat. Add onion, garlic, and broccoli. Cook slowly, stirring occasionally, until vegetables are soft. Spoon vegetables into crust and sprinkle with cheese.
Combine eggs and milk. Season with salt and pepper. Stir in remaining 1 tablespoon melted butter. Pour egg mixture over vegetables and cheese.
Bake in preheated oven until center has set, 30 to 50 minutes.