Stir together the mayonnaise, mustard, caraway, 1/4 teaspoon of salt and several grinds of black pepper in a small bowl and set aside.
Heat a 2-burner griddle over medium heat. Melt 1 tablespoon of the butter over the surface of the griddle. Add the onions and peppers and season with 1/4 teaspoon salt. Cook, tossing occasionally, until tender and golden, 8 to 10 minutes. Push to one side of the griddle and Scrape the griddle clean.
Combine the beef, Worcestershire and 3/4 teaspoon salt in a medium bowl and mix with your hands to combine. Form into 2 oval patties about 1/2 inch thick each, slightly larger than the size of the slices of toast (they will shrink a little as they cook). Season both sides of the patties with the remaining 1/4 teaspoon of salt and a couple cranks of black pepper.
Melt the remaining tablespoon of butter on the grill over medium to medium-high heat. Add the patties on one side of the griddle and press to flatten with a metal spatula. Cook, flipping once, until browned and almost cooked through, 2 to 3 minutes per side. Meanwhile, lay the Texas toast down on the other side of the griddle. Toast one side, 1 to 2 minutes, then remove to your cutting board.
To assemble, spread the toasted side of 2 pieces of bread with half of the mayo mixture. Add a slice of Swiss cheese to each, then a burger patty. Top with the onion mixture and the American cheese. Spread the remaining mayo mixture on the toasted sides of the last 2 pieces of Texas Toast and top the sandwiches mayo-side down.
Set the sandwiches on the griddle over medium low heat. Toast, flipping once, until the cheese is melted and the bread is crisp and toasty, about 3 minutes per side. Cut in half to serve, with ketchup on the side.