Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add apples, cover, and steam until tender, about 15 minutes.
Mix softened apples, honey, brown sugar, and cinnamon together in a bowl.
Spread puff pastry squares on a flat work surface. Place 1 tablespoon of the apple mixture in the middle of each square. Fold the corners of each square upward and pinch the edges together to make sure the filling stays in place. Put the squares on a baking sheet.
Bake in the preheated oven until the pastry rises and turns golden brown, 20 to 25 minutes.
