Preheat the oven to 500°F. Line a rimmed baking sheet with parchment paper.
Divide the ground beef into 4 equal portions and form each into a 4-inch square patty, keeping an even thickness. Press your thumb into the center of each to make a dent (this prevents the burgers from puffing up in the middle and cooking unevenly). Season with salt and pepper.
Place a large cast-iron skillet over medium-high heat. When it's really hot (just beginning to smoke), add the burgers and cook for 1-1/2 to 2 minutes per side. (Don't worry about cooking the burgers all the way through; you just want them to have a browned crust.) Immediately remove them from the skillet and set them aside on a plate to cool slightly.
Divide the pizza dough into 4 equal portions and roll each one out to a rectangle shape, 5 or 6 inches wide and 8 to 10 inches long.
Place a burger on one side of each rectangle of dough. Top each with a few onion slices (if using), 3 pickle slices, and 1 slice of cheese. Sprinkle with salt and pepper. Fold the exposed dough over the burger. Wet your fingers with a little water and twist/crimp the dough to seal. Using a sharp knife, cut vents in the top of each calzone.
In a small bowl, whisk the egg with a splash of water. Brush this egg wash over the tops of the sealed calzones, then transfer the calzones to the prepared baking sheet.
Bake for 10 minutes, or until the dough is puffed and golden. Remove from the oven and set aside to cool slightly while you make the sauce.