Make the slaw: Put the cabbage in a small bowl. Sprinkle the vinegar, sugar, salt and pepper to taste, and the celery seeds over the top and toss well. Let stand, tossing frequently, until the cabbage wilts, about 20 minutes.
Make the sandwiches: In a small bowl, stir together the mayonnaise, ketchup, relish and paprika. Add half of the dressing to the lobster meat and toss well to combine. Arrange the bread slices on a work surface and spread a little of the remaining dressing on each slice. Put a slice of cheese on each piece of bread; divide the lobster meat among 4 of the slices, then top each with one-quarter of the slaw. Invert the remaining cheese-covered bread slices onto each sandwich. Spread 1/2 tablespoon butter on the top of each sandwich.
Melt the remaining 1 tablespoon butter in a large nonstick skillet over medium heat and swirl the pan to coat it. Arrange the sandwiches, butter-side up, in the pan and cook until the bottoms are golden, about 5 minutes. Carefully flip the sandwiches and cook until the other sides are golden brown and the cheese is melted, about 5 more minutes. Remove the sandwiches from the pan and let stand 1 minute before slicing on the diagonal and serving warm.
To cook your own lobster, boil two 2 1/4-pound lobsters in a large pot of salted water, covered, 5 minutes. Uncover, reduce the heat and simmer 8 more minutes. Drain and let cool; remove the meat.
Photograph by Con Poulos
Recipe from Endless Summer Cookbook. Copyright (c) by Katie Lee. Published by Stewart, Tabori & Chang, an imprint of Abrams