Step 1Heat 2 tablespoons of butter in a 10-inch nonstick skillet over medium-high heat. When the butter is sizzling, add the onion and cook, stirring often, until softened but not brown, about 4 minutes. Add the cabbage and vinegar. Cook gently, covered, until the cabbage is tender but still textured, about 12 minutes. Add 2 tablespoons of sugar, the caraway seeds, salt and pepper to taste. Cook gently 5 minutes longer, stirring often to prevent burning.
Step 2Toss the apple slices with the remaining 1 1/2 teaspoons of sugar in a large bowl. Heat 1 tablespoon of butter in an 8-inch nonstick skillet over medium-high heat. When the butter starts to brown, add the apple. Cook until golden and tender but still intact, about 6 to 8 minutes, stirring often.
Step 3To assemble, generously spread Dijon mustard on 4 bread slices. Cover the mustard with the warm apple. Cover the apples with a cheese slice. Divide the warm cabbage equally among the remaining 4 slices of bread. Place a cheese slice on the cabbage. Leave the sandwiches open-faced for cooking; then close after cooking.
Step 4Heat the oven to 250 degrees.
Step 5Heat the griddle over medium heat and lightly butter. Place 2 open-faced sandwiches on the griddle. Cook, tented with foil, until the cheese is melted, about 4 minutes. Close the sandwich, pressing it lightly together. Turn it over and place on a baking sheet. Keep warm in the oven while cooking the remaining sandwiches. Serve warm.