Preheat oven to 350 degrees F (175 degrees C). Press pie pastry into an 8-inch pie dish.
Heat a skillet over medium heat; cook and stir bacon for 3 minutes. Add onion, spinach, and courgette to bacon; cook and stir until onion and courgette are tender, spinach is wilted, and bacon is fully cooked, 5 to 10 minutes. Transfer bacon-courgette mixture to the pie pastry.
Beat milk, eggs, salt, and pepper together in a bowl; pour into pastry crust. Top egg mixture with mozzarella cheese.
Bake in the preheated oven until cheese is browned and quiche bounces back when touched, about 35 minutes.