Preheat your oven to 400°F (200°C).

Dust your work surface with flour, and roll out the puff pastry sheets until they are about 1/8-inch thick.

Using a sharp knife or pizza cutter, slice each pastry sheet into rectangles, roughly 3 inches wide and 5 inches long.

On one half of each rectangle, layer a slice of cheese followed by a slice of ham, and then top with another slice of cheese.

Fold the pastry over the filling to create a pocket. Press the edges together gently to seal, and use a fork to crimp the edges for a decorative touch.

Brush each pocket with beaten egg to promote a golden color while baking.

Place the sticks on a baking sheet lined with parchment paper.

Bake in the preheated oven for 15-20 minutes or until golden and crispy.

Let the sticks cool slightly, then serve them hot with your favorite dipping sauce, such as marinara or mustard.