Preheat the broiler.
Grate the shallot over a large bowl. Add the meat loaf mix, breadcrumbs, half the ketchup, egg, Worcestershire sauce, 1 teaspoon salt and several grinds of pepper. Use your hands to thoroughly combine the mixture, then divide it into 4 equal balls and shape each into a patty a little less than 1/2 inch thick. Arrange on a baking sheet and broil until the tops of the patties begin to firm up and brown, 3 to 4 minutes. Flip the patties with a metal spatula and continue to broil until cooked through, 2 to 3 minutes more.
Brush each patty with some of the remaining ketchup and top each with a slice of cheese. Continue to broil until the cheese melts, about 30 seconds more.
While the patties broil, put the green beans, butter, 1 tablespoon water, 1/2 teaspoon salt and a few grinds of pepper in a medium microwave-safe bowl. Cover and microwave until hot, 4 to 5 minutes.
Open the buns and spread the cut sides with mayonnaise. Place a lettuce leaf and patty in each. Serve each sandwich with a small bowl of green beans.
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