How to Make It
Heat oven to 425° F. Dividing evenly, on 2 rimmed baking sheets, toss the potatoes with the barbecue spice rub and 2 tablespoons of the oil. Roast, tossing once, until browned and crisp, 25 to 30 minutes.
Meanwhile, in a small bowl, mix together the mayonnaise, relish, and ketchup.
Form the beef into 4 patties and place each between 2 sheets of wax paper; press to a ½-inch thickness. Heat the remaining teaspoon of oil in a large skillet over high heat. Season the burgers with ½ teaspoon salt and ¼ teaspoon pepper and cook to the desired doneness, 2 to 3 minutes per side for medium-rare.
Form sandwiches with the buns, burgers, mayonnaise sauce, lettuce, tomato, and onion. Serve with the fries.