Butter, for greasing the pie dish
5 medium Granny Smith apples, peeled, cored and quartered (about 2 1/2 pounds)
1 cup coarsely grated sharp white Cheddar (4 ounces)
2/3 cup dark brown sugar
2 tablespoons unsalted butter, cut into 1/2-inch pieces, at room temperature
1 tablespoon all-purpose flour
1 teaspoon ground cinnamon
1/4 teaspoon freshly ground nutmeg
1/4 teaspoon kosher salt
Two 9-inch-diameter, unroll-and-bake refrigerated pie crusts
1 egg white, beaten until frothy