Preheat the oven to 375°F. Arrange the puff pastry on a generously floured surface and roll it out an additional 2 inches or so on all sides.

Starting from one of the short sides of the pastry, cut the pastry into four equal-sized long strips, then cut each strip in half lengthwise so you have 8 equal-sized rectangles. Place the strips on a large parchment-lined baking sheet, leaving a few inches between pieces, and chill for 10 minutes. Combine the mustard and relish in a small bowl and spread 1 tablespoon of the mixture on four of the strips, leaving a ½-inch border around the edges.

Divide the ham and cheese evenly among the mustard-topped pastry pieces, making sure to leave the same ½-inch border around the edges. Arrange the four remaining pastry halves on top of the mustard-topped halves, then use a fork to crimp them around the edges to seal. Brush each pocket with egg, then use a small paring knife to cut a few small slits on the top of each pocket.

Bake until golden brown and shiny on top, 20 to 25 minutes. Serve with the remaining mustard-pickle spread.