Wash and peel the potatoes. Cut them into thin strips to create fries. Soak the cut potatoes in cold water for at least 30 minutes to remove excess starch. Drain and pat dry with paper towels.

In a medium bowl, combine mayonnaise, ketchup, Dijon mustard, pickle relish, garlic powder, onion powder, salt, and pepper. Mix well and set aside.

In a deep fryer or a heavy pot, heat vegetable oil to 350°F (175°C). Carefully add the dried potato strips in batches, frying until golden brown and crispy, about 5-7 minutes. Remove with a slotted spoon and drain on paper towels. Season with salt while hot.

Preheat a grill or skillet over medium-high heat. Form the ground beef into 4 equal patties. Season both sides with salt and pepper. Cook the patties on the grill or skillet for about 4-5 minutes on one side. Flip the burgers, place a slice of cheddar cheese on each patty, and cook for an additional 4 minutes, or until they reach your desired doneness.

Toast the burger buns lightly on the grill or in a toaster. Spread a generous amount of the tangy burger sauce on the bottom half of each bun. Add a lettuce leaf, followed by the cooked burger patty with melted cheese, then layer with sliced tomato, pickles, and red onion. Top with the other half of the bun.

Plate the cheeseburgers alongside a generous serving of golden fries. Drizzle extra tangy burger sauce on the side for dipping, if desired.