Preheat the oven to 350 degrees F (175 degrees C).
Heat oil in an oven-safe skillet over medium heat and cook garlic until it starts to sizzle, about 1 minute. Add asparagus and mushrooms and cook until softened, 4 to 5 minutes. Season with salt and pour in white wine. Discard garlic.
Mix eggs, milk, 2 tablespoons Parmesan cheese, parsley, salt, and pepper in a bowl. Pour into the skillet over the asparagus and mushrooms.; stir gently. Cook until eggs start to set, about 3 minutes. Sprinkle with remaining 1 tablespoon Parmesan cheese and transfer to the oven.
Bake in the preheated oven until frittata has risen and is set, about 20 minutes. Cool slightly before serving.