Preheat the oven to 375 degrees F (190 degrees C).
Heat olive oil in a skillet over medium heat. Add mushrooms, onion, and garlic; cook and stir until lightly browned and tender, about 5 minutes. Add spinach and cook until wilted, about 3 minutes.
Whisk eggs, milk, salt, pepper, and nutmeg together in a small bowl.
Line a 9-inch pie plate with pie crust. Spoon spinach mixture into the crust. Pour egg mixture on top. Sprinkle Swiss cheese over the top.
Bake in the preheated oven until the center is set, 35 to 40 minutes.
