Preheat the oven to 400 °F (200 °C). Grease a 12-cup muffin pan and set aside. Peel, core, and finely dice the apples.
In a medium saucepan over medium heat, melt the butter. Once melted, add the chopped apples, maple syrup, cinnamon, and nutmeg. Stir until fully coated. Cook the apples, stirring often, until softened, about 10 minutes.
In a small bowl, stir together the cornstarch and 1 tablespoon of cold water until smooth. Pour this cornstarch mixture into the apples, stir, and cook until thickened. Remove from the heat to cool.
Unroll each sheet of puff pastry and slice it into 6 equal rectangles to form 12 rectangles total. Press each rectangle into a muffin cup.
Spoon about 2 tablespoons of the cooled apple filling into each pastry cup. Fold the corners inward and pinch them together to slightly close.
Brush the top of each pastry with the whisked egg (or milk) and sprinkle with coarse brown sugar, if desired.
Bake the croissants for 20–25 minutes or until golden. Allow to cool briefly in the muffin pan before serving.
