How to Make It
Preheat the oven to 400°. Line a rimmed baking sheet with parchment paper. On a lightly floured work surface, gently roll out the puff pastry to a roughly 13-inch square 1/16 inch thick. Cut out four 6-inch squares and transfer them to the prepared baking sheet. Refrigerate the pastry for 5 minutes.
Meanwhile, lay the ham slices on a work surface. Spoon 1/4 cup of the shredded Gruyère on one side of each slice of ham, then fold the other side of the ham over the cheese to form a neat package.
Lightly brush the edges of the pastry squares with water. Set the ham-and-cheese packages on the squares and fold the pastry corner to corner to form triangles. Press the edges of the turnovers together and crimp decoratively with a fork. Brush the turnovers with the beaten egg and bake them for about 30 minutes, or until golden brown and puffed. Let cool slightly and serve.