Wild Mushroom Quiche with Walnut Crust

Name: Wild Mushroom Quiche with Walnut Crust
Ingredients: 1 1/2 cups walnuts; 1 1/3 cups all-purpose flour; 2 tablespoons sugar; 1/2 teaspoon fine salt; 1 stick unsalted butter, cut into 1-inch pieces; 3 tablespoons olive oil; 1 pound mixed wild mushrooms, such as oyster, shiitake, and trumpet; 3/4 pound white or cremini mushrooms, thinly sliced; Sea salt and freshly ground pepper; 3 tablespoons unsalted butter; 2 shallots, minced; 1 tablespoon chopped fresh thyme; 3 1/3 cups grated Parmesan; 1/2 cup shredded Comte or Mahon cheese; 3 large eggs; 1 cup whole milk; 3/4 cup heavy cream; 2 tablespoons chopped fresh flat-leaf parsley, for garnish; Crisp greens, for serving