Cheddar Crust Apple Pie

Name: Cheddar Crust Apple Pie
Ingredients: 1 1/4 cups unbleached all-purpose flour; 1/4 cup cake flour; 1/4 teaspoon salt; 1 stick unsalted butter, frozen solid, slightly thawed; 2 tablespoons lard or vegetable shortening, frozen solid, slightly thawed; 1 cup grated extra-sharp Cheddar cheese; 2 teaspoons white vinegar mixed with 1/3 cup very cold water; 6 to 7 large Delicious apples (3 pounds), peeled, cored and sliced thinly into 1/16-inch thick slices; 2 tablespoons freshly-squeezed lemon juice; 1 cup sugar, plus additional for sprinkling; 1/3 cup unbleached all-purpose flour; 3/4 teaspoon cinnamon; 3/8 teaspoon salt; 2 tablespoons unsalted butter, softened; 1 egg beaten with 1 tablespoon cold water, for glaze