Preheat the oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Remove dough from the can and roll out into a 9x16-inch rectangle. Cut into 4 equal rectangles.
Place 1 rectangle horizontally on a work surface. Brush mustard over the surface, leaving a 1/2-inch border. Arrange ham, pork, salami, Swiss cheese, and 2 pickle chips, in that order, on the left side of the rectangle. Leave the border and right side of the dough exposed.
Whisk egg and milk together to make an egg wash. Brush around the border. Fold the right side over the filling and press to seal; crimp the edges with a fork. Cut a few small slits in the top of the hand pie to vent. Brush some egg wash over the top.
Repeat with remaining dough, mustard, filling, and egg wash. Arrange hand pies on the prepared baking sheet.
Bake in the preheated oven until crisp and golden brown, 22 to 25 minutes. Cover with foil if starting to brown too quickly. Remove pies to a rack to cool for 5 minutes.
