Melt butter in a large skillet over medium heat; cook and stir green onions and garlic until green onions are softened, about 5 minutes. Cook and stir turkey bacon in the green onion mixture until fragrant.
Stir spinach, shiitake mushrooms, and broccoli into green onion mixture; cook and stir vegetables until softened, 5 to 7 minutes. Mix in 1/4 cup Cheddar cheese.
Spoon vegetable mixture into the pie crust.
Beat eggs, milk, sea salt, and black pepper together in a bowl; pour egg mixture gently over vegetable filling, slightly stirring eggs into the filling to combine.
Bake in the preheated oven for 15 minutes; sprinkle 1/2 cup Cheddar cheese over the top of the quiche and bake 10 additional minutes. Cover quiche with aluminum foil and bake until quiche has set and a toothpick inserted into the quiche comes out clean, about 10 more minutes.
Allow quiche to cool for 5 minutes before serving.