1. For the crust: Preheat the oven to 350 degrees F. Pulse together the walnuts, flour, sugar, and salt to a fine meal in a food processor. Pulse in the butter until it forms a crumbly dough. Press into the bottom and up the sides of a 9-inch springform pan. Place on a baking sheet and bake until lightly browned, 15 to 20 minutes. Remove and let cool.
2. For the filling: Reduce the oven temperature to 325 degrees F. Heat the olive oil in a large skillet over high heat. Add the wild mushrooms, white mushrooms, and season with salt and pepper. Cook, stirring occasionally, until soft, about 5 minutes. Lower the heat to medium; add the butter, shallots, and thyme. Cook, stirring frequently, until the mushrooms are tender, about 10 minutes more. Adjust the seasoning and set aside.
3. Combine the Parmesan and Comte cheeses in a small bowl and set aside. Scatter half the mushrooms and half the cheese evenly over the crust. In a large bowl, whisk together the eggs, milk, and heavy cream until smooth and frothy. Pour half the egg mixture over the mushrooms. Repeat with the remaining mushrooms and egg mixture and top with the remaining cheese.
4. Bake the quiche until golden brown on top and just set with a little wiggle in the very center, 50 to 60 minutes. Run a clean paring knife between the crust and the pan to separate. Let cool slightly, about 25 minutes. Garnish the quiche with the parsley. Carefully remove the springform pan ring. Cut the quiche into wedges and serve warm or at room temperature with crisp greens.
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