1. Make the crust: In a large bowl, whisk together flour and salt. Cut in cold butter until mixture resembles coarse meal. Add ice water, one tablespoon at a time, stirring with a fork until dough comes together. Shape into a disc, wrap in plastic, and refrigerate for at least 30 minutes.
2. Prepare the filling: Heat olive oil in a large skillet over medium heat. Cook onion until softened, about 4 minutes. Add mushrooms and cook until golden, 6-8 minutes. Stir in garlic and cook for 30 seconds. Add spinach and cook until wilted. Remove from heat and let cool slightly.
3. Preheat oven to 375 degrees F. Roll out chilled dough on a floured surface to a 12-inch circle. Transfer to a 9-inch pie plate, pressing into bottom and sides. Trim and crimp edges. Line with parchment paper and fill with pie weights. Bake for 15 minutes, then remove parchment and weights and bake for 5-7 minutes more, until lightly golden.
4. In a medium bowl, whisk together eggs, milk, heavy cream, salt, pepper, and nutmeg until smooth.
5. Spread the cooled spinach and mushroom mixture over the pre-baked crust. Sprinkle cheese on top. Pour the egg custard over the filling.
6. Bake for 35-40 minutes, or until center is set and a knife inserted near the center comes out clean. Let cool on a wire rack for at least 15 minutes before slicing.