My parents owned a fifties-style diner called Chubby's when I was a kid, so a patty melt was my introduction to rye bread. When I was studying for my baking degree, one of my favorite things to make was rye bread. If you don't yet appreciate the amazing flavor of caraway seed, it's one of those things that will sneak up on you and make you fall in love with it sooner or later.
For the Thousand Island dressing: Stir together all the ingredients in a small bowl until well combined. Set aside.
For the universal tuna salad: Open the tuna cans and, using the back of a spoon, press as much of the liquid out as you can. Whisk together the mayonnaise, lemon juice, olive oil, mustard, salt and pepper in a small bowl. Fold in the tuna and stir until incorporated.
For the patty melt: Stir together the tuna salad and the sauerkraut in a medium bowl, and divide the mixture into 4 equal portions. Using your hands, form each portion into a ball, and squeeze out any remaining liquid.
Heat the oil in a medium skillet over medium-high heat. Add the balls of tuna, pressing them into flat patties using a spatula. (It's okay if they run into each other.) Cook until caramelized on the bottom, 4 to 5 minutes. Gently turn the patties over, and cook until the other side is caramelized. Transfer the patties to a plate, and cover with aluminum foil to keep warm.
Spread 2 slices of the bread with 1/2 teaspoon butter on each side, and put them in the skillet over medium-low heat. Top each bread slice with a Swiss cheese slice, and cook until the cheese melts, about 1 minute. Spread 1 tablespoon of the dressing on each cheese slice. Top each with a warm tuna patty and 1 Swiss cheese slice. Top each sandwich with another slice of bread, and butter the outside of each bread slice with 1/2 teaspoon butter. Cook for 1 minute; turn and cook until the bread is toasted and the cheese is melted, another 1 to 2 minutes. Repeat the process to make 2 more patty melt sandwiches. Serve the sandwiches hot with the remaining dressing.
Serve the extra Thousand Island Dressing on the side to dip the sandwiches into, or squeeze a swipe of Sriracha chili sauce down the middle of each sandwich.