Preheat the oven to 400° F.
Heat pan on medium heat and add oil. When oil is hot, add mushrooms. Season with salt and pepper, 5 to 8 minutes, until water from the mushrooms evaporates. Set mushrooms aside.
While the mushrooms are sautéing, prepare the leeks. Cut off the roots and dark green parts; only use the white and light green parts. Starting ½ inch from the base, slice each leak all the way to the top, rotate the leak 90 degrees and cut again, making a plus sign. Run leeks under water to get all the dirt out from between the many layers. Once washed, cut in half lengthwise then slice into ⅛-inch pieces.
Add oil to the same pan used for the mushrooms. Return to medium heat. Add leeks. Let them sweat and get translucent. Set leeks aside.
Next, prepare the filling. Crack 3 eggs into a large bowl or measuring cup and beat like you would for an omelet. Add milk, heavy cream, nutmeg, salt and pepper. Mix well.
If using a store-bought crust, defrost for 10 minutes. Lightly press any cracks to seal them so the liquid does not seep through.
Using a fork, prick holes in the bottom and side of the crust. Cook for 6 minutes.
While the crust is cooking, beat the fourth egg in a bowl for an egg wash. At the end of the 6 minutes, remove the crust from the oven.
Using a basting brush, paint the egg wash on the crust, covering it all. (This seals the crust and keeps it crunchy after it is filled.) Cook the crust for another 2 to 3 minutes then let it cool.
Lower the oven temperature to 375° F. Fill the crust. Line the base with the shredded cheese. Add the leeks and mushrooms and spread them out evenly. Pour the egg mixture over. Top with some additional shredded cheese. Cook for 35 to 45 minutes.
The quiche will look like a soufflé and start to get golden. Quiche is done when a knife inserted in the center comes out dry.