Divide the meat into 4 equal portions (about 6 ounces each). Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb. Season both sides of each burger with salt and pepper. Heat the oil in the pan or griddle over high heat until the oil begins to shimmer. Cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey. Flip over the burgers. Cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese) or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side. Add the cheese, if using, to the tops of the burgers during the last minute of cooking and top with a basting cover, close the grill cover, or tent the burgers with aluminum foil to melt the cheese.
Combine the mayonnaise and roasted garlic in a small bowl and season with salt and pepper to taste. Spread both sides of each bun with the mayonnaise and the mustard. Place a slice of cheese on each bun bottom, place a burger on top, and then top the burger with a slice of ham, another slice of cheese, and some pickle slices. Cover with the bun tops.
Cook the burgers on a sandwich press or wrap the burgers in aluminum foil and cook in a hot skillet over high heat (put a heavy skillet on top of the burgers to press them) until golden brown and the cheese has melted, about 1 1/2 minutes per side. Serve immediately.