Preheat oven to 400°. Remove pastry sheet from package and let thaw (about 10 minutes).
While pastry thaws, core apples and cut in half. Thinly slice apples on a mandolin, in a food processor, or by hand. Put apples in a bowl.
Melt butter in a small pan over moderate heat, swirling pan, until golden brown, about 2 minutes. Pour butter over apples and toss.
Place thawed pastry on a sheet of parchment cut to fit a large baking sheet. Gently roll pastry into a 15 x 12-inch rectangle on the parchment. Lift parchment and transfer it with the pastry to the baking sheet. Sprinkle bread crumbs over pastry.
Stir sugar into apples, then spread apples over pastry, leaving a 1-inch border all around. Fold edges of pastry over apples, pressing firmly on corners and sides to seal.
Bake until apples are tender, 30 to 35 minutes. Sprinkle cheese over apples and bake until cheese is bubbling, 5 to 7 minutes more. Serve warm.