When assembling the sandwiches, coat each piece of bread with a slathering of sweet butter.
Bring a large pot of water to a boil, and add potatoes and 1 tablespoon salt. Return to a boil. Reduce heat; simmer until tender, about 7 minutes. Drain. Let cool.
Spread bread slices with butter; sprinkle each with 2 teaspoons cheese. Divide potatoes among bread. Top each with apple slices, and sprinkle with 1 teaspoon cheese. Top with onion slices, dividing evenly. Sprinkle with salt, and drizzle each with 1 teaspoon pickling liquid.