Preheat the oven to 350 degrees F with a rack positioned in the middle of the oven.
Whisk together the mayonnaise, ketchup, Dijon mustard and sweet pickle relish in a small mixing bowl. Season to taste with kosher salt. Reserve.
Use a serrated knife to trim away the crust from 8 of the rye toast squares. Use a 2-inch round cutter to cut the remaining rye toast squares into rounds. Spread a thin layer of salted butter on 1 side of each rye toast. Place the buttered rye toasts on a baking sheet, buttered-side down. Spread 1 teaspoon of the reserved dressing on each piece of rye toast. Top each with 1 folded piece of corned beef and 2 pieces of Swiss cheese. Place in the preheated oven and bake until the corned beef is heated through and the cheese is melted, 3 to 5 minutes.
Remove to a large platter and top each with a heaping teaspoon of sauerkraut, a dill pickle chip and a sprinkle of crushed kettle-style potato chips.