Dough
1 medium potato, peeled and cut into large chunks
1/4 cup unsalted butter
1 tablespoon active dry yeast
1/2 cup packed brown sugar
1 large egg
1 teaspoon salt
5 to 5 1/2 cups all-purpose or bread flour

Filling
6 tablespoons butter, at room temperature and divided
3 cups diced apple
1/4 teaspoon salt
1 1/2 cups packed brown sugar, divided
1 1/2 tablespoons ground cinnamon, divided

caramel sauce
1 1/8 cup granulated sugar
2 1/2 tablespoons water
1/4 teaspoon lemon juice
1 tablespoon corn syrup
3/4 cups heavy cream
2 tablespoons butter
1/4 teaspoon salt