Lay each piece of puff pastry on its own piece of parchment paper and roll them lightly to remove the fold lines. Preheat a skillet to medium-high. Saute the apples and shallots in butter for 3-5 minutes. Add the nutmeg, then salt and pepper to taste.
Top both pieces of puff pastry with apple mixture and distribute evenly. Then top them with shredded cheese.
Fold the pastry sheets inward 1/4 of the width on both ends. Fold in again so that the dough meets in the middle. Fold one more time. Wrap the rolls in the paper and place the rolls in the fridge for at least 30 minutes to chill before cutting. (Wrap them well if you are prepping ahead.)
When ready to bake, preheat the oven to 450 degrees F. Carefully slice each roll into 12-15 1/2 inch slices and lay them flat on parchment paper lined baking sheets. Whisk the egg and brush it over the tops. Bake for 12-14 minutes.