Step 1 Heat the oven to 375 degrees.
Step 2 Coarsely chop the spinach. In a large skillet over medium heat, melt 1 tablespoon butter. Add the spinach and cook until the water evaporates, about 10 minutes, stirring occasionally. Transfer the spinach to a medium bowl and set aside.
Step 3In a large, heavy-bottom saucepan over high heat, saute the leeks, mushrooms and garlic with the remaining butter until the garlic is aromatic, about 2 minutes. Add the wine, reduce the heat to medium and cook until the vegetables are softened and the liquid evaporates, about 8 to 10 minutes. Transfer the mixture to a separate medium bowl and set aside.
Step 4In a large mixing bowl, whisk together 2 eggs, the ricotta, goat cheese, 1 tablespoon thyme, the parsley, nutmeg, lemon juice and Parmesan. Season to taste with salt and pepper.
Step 5Divide the chilled dough into two equal portions. On a lightly floured surface, roll the halves into two round sheets approximately 12 inches in diameter. Drape and press one round into a 9-inch pie plate (there will be some overhang). Set the remaining round on plastic wrap and reserve.
Step 6Spoon half the cheese mixture into the pie plate, then cover with half the spinach. Sprinkle with half of the remaining thyme. Cover the spinach with half of the mushroom leek mixture. Cover with the remaining spinach, then the remaining mushroom and leek mixture. Sprinkle over the remaining thyme, then spoon over the remaining cheese mixture.
Step 7Place the second sheet of dough over the top of the pie, then trim and crimp the edges. Slice a few small steam vents in the surface. Lightly beat the remaining egg and brush gently over the top of the pie. Bake until golden brown, about 1 hour. Rotate halfway through baking for even coloring.