1. Preheat the oven to 350 degrees F (175 degrees C).
2. Melt butter(3 tbsp) in a large saucepan over medium-low heat. Add chopped broccoli(2 cups), minced onion(1), and garlic gloves(2); cook, stirring occasionally, until vegetables are soft. Spoon vegetables into 9 inch pie crust(1) and sprinkle with shredded mozzarella cheese(1.5 cups); set quiche on a baking sheet.
3. Combine eggs(4) and milk(1.5 cups) together in a bowl; season with salt and pepper. Stir in remaining 1 tablespoon melted butter; pour egg mixture over vegetables.
4. Bake in preheated oven until the center of the quiche has set, about 30 to 50 minutes.