For the slaw: Put the cabbage in a bowl, sprinkle the vinegar, sugar, celery seeds and salt and pepper to taste over the top and toss well. Let stand, tossing frequently, until the cabbage wilts, about 20 minutes.
For the sandwiches: In a small bowl, stir together the mayo, ketchup, relish and paprika. Spread the dressing on one side of each slice of bread. Put a slice of cheese on each piece of bread; divide the meat among 4 of the slices. Top each sandwich with some slaw and invert the remaining cheese-covered bread slices onto each sandwich. Spread 1/2 tablespoon of the butter on the top of each sandwich.
Melt the remaining 1 tablespoon butter in a large nonstick skillet or griddle over medium heat. Arrange the sandwiches buttered-side up in the skillet and cook until the bottoms are golden, about 4 minutes. Carefully flip the sandwiches and cook until the undersides are golden brown and the cheese is melted, about another 4 minutes. Slice the sandwiches diagonally and serve.