For the tots: Preheat the oven to 400 degrees F. Line a small baking sheet with aluminum foil and mist with nonstick cooking spray. Spread the Cheddar in a single layer on the baking sheet. Bake until bubbly and lightly browned, 16 to 18 minutes. Cool completely on the tray, and then coarsely chop.
Meanwhile, poke the potatoes with a fork and microwave until softened but not completely cooked through, 6 to 8 minutes. When cool enough to handle, peel and discard the skin. Grate the potatoes on the large holes of a box grater. Add the chopped Cheddar, salt, pepper and garlic powder. Stir with a fork to combine everything, and then stir in the flour.
Heat about 3 inches of oil to 350 degrees F in a heavy pot or Dutch oven (or set a countertop fryer to 350 degrees F). Using a tablespoon of potato mixture, form each tot into a squat cylinder about 1 1/2-inches in diameter and 3/4-inch tall. Fry in small batches until golden and crisp on the outside, 1 1/2 to 2 1/2 minutes. Drain on a paper-towel-lined plate or baking sheet and sprinkle with salt while warm. Serve with the burgers, or freeze for later use in self-sealing freezer bags, and reheat in a 375 degree F oven for 7 to 10 minutes.
For the sauce: Mix the mayonnaise with the relish, capers and a pinch each of salt and pepper. Set aside.
For the burger: Divide the meat into 4 equal portions and form into 4-inch patties about 1/4-inch thick. Sprinkle all the burgers with salt and pepper. Heat a large skillet or a griddle over medium-high heat. Add 1 tablespoon of the butter and, when just melted, add the burgers. Cook, turning once, until browned and cooked to medium-well, 2 to 3 minutes on each side.
Meanwhile, spread the buns with the remaining tablespoon of butter and toast in the oven, 3 to 5 minutes.
To build the burgers, evenly spread the sauce on the bottom half of the buns. Evenly top with the shredded lettuce and onion rings. Add a burger patty, ketchup and mustard, and top with the bun. Serve with the Cheddar tots.