Preheat oven to 350 degrees. Heat olive oil in a pan. Saute garlic, onion, and zucchini until tender, 6 to 8 minutes. Let cool.
Whisk eggs, milk, thyme, salt, and black pepper. Spread cooked vegetables on a baked 9-inch pie crust.
Add goat cheese, then baby spinach. Slowly pour in egg mixture and sprinkle with grated Parmesan and more pepper, if desired.
Bake on a rimmed baking sheet 50 to 60 minutes until the center is just set. Let cool 30 minutes before serving.