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 a gas grill to high or heat coals in a charcoal grill until they glow bright orange and ash over.
Brush the burgers with the oil. Grill 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 or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side. Remove the burgers to a plate.
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.