Apple and Cheese Pie

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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