Wild Mushroom Quiche with Walnut Crust

Name: Wild Mushroom Quiche with Walnut Crust
Ingredients: 1 1/2 cups walnuts; 1 cup all-purpose flour; 2 tablespoons sugar; 1/2 teaspoon fine salt; 5 tablespoons unsalted butter, cut into 1-inch pieces; 2 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; 2 tablespoons unsalted butter; 2 shallots, minced; 1 tablespoon chopped thyme; 2 1/3 cups freshly grated Parmesan (4 ounces); 1/2 cup shredded Comte or Mahon cheese; 3 large eggs; 1 1/4 cup heavy cream; 2 tablespoons chopped flat-leaf parsley, for garnish; Crisp greens, for serving