Position an oven rack on the bottom third of the oven and preheat to 400 degrees F. Butter the sides and bottom of a 9-inch-diameter glass or ceramic pie dish.
Cut the apple quarters into 3 wedges. Mix the apples, cheese, sugar, butter, flour, cinnamon, nutmeg and salt in a medium bowl.
Unroll one crust and place in the buttered pie dish. Brush the inside of the crust with some of the egg white to coat. Pour the apple filling into the crust. Unroll the second crust and place on top of the filling. Pinch the edge of the top crust and the edge of the bottom crust together to seal. Fold the double edge under and crimp, making a decorative edge. Brush the top of the pie with more of the egg white. Cut several slits in the top crust to allow steam to escape.
Bake the pie until the crust is deep golden and the filling is bubbling through the slits, about 1 hour. Cover the edges of the pie with foil if the crust browns too quickly. Cool the pie for about 20 minutes before serving.