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.
Place the burgers on the bun bottoms and, if desired, top with tomato, lettuce, onion, and a dollop of horseradish mustard mayonnaise. Pile on the potato chips, top with the bun tops, and serve immediately.
Whisk together the mayonnaise, mustard, and horseradish in a small bowl and season with salt and pepper. Cover and refrigerate for at least 30 minutes to allow the flavors to meld. The sauce can be prepared 1 day in advance and kept covered in the refrigerator.
Toasted Burger Buns I think the taste and the texture of buns are best when lightly toasted. To toast a bun on a grill, grill pan, or griddle, split the bun open, place it cut side down on the grill, and grill until light golden brown, about 10 seconds. Alternatively, you can place the split bun halves cut side up on a baking sheet and cook them under a preheated broiler until light golden brown, about 30 seconds. Keep a close eye on them and don't let them get too brown or else that texture will take on a life of its own and defeat the purpose of starting with a soft bun. *You would typically find this sauce served alongside a thick cut of prime rib. The blend of mellow mayonnaise, pungent horseradish, and sharp Dijon mustard complements steak so well that it is no surprise that it does the same for a beef burger. A rich salmon burger would also take beautifully to the sauce. For a touch of extra flavor and color, try adding a few tablespoons of chopped fresh herbs such as chives, dill, or tarragon.