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puree the mayonnaise, ketchup, mustard, pickles and brine, sugar and cayenne in a mini food processor or blender until smooth.
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transfer to a small bowl and refrigerate until ready to serve.
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form the beef into 4 pucks, about 2 inches wide and 1 1/2 inches thick.
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melt the butter on a griddle or large skillet over medium heat. open the buns and place open-side down on the griddle.
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cook until lightly toasted, about 2 minutes.
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remove to a plate.
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wipe off the griddle.
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increase the heat to medium high and brush with the vegetable oil.
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working in two batches, transfer the patties to the griddle, about 3 inches apart, and press firmly with a large spatula (or two small spatulas) to flatten into very thin patties.
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cook, pressing the edges of the patties, about 20 seconds.
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season each patty generously with salt and black pepper.
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continue cooking until the edges are crisp and the beef is browned, 1 to 1 1/2 more minutes.
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flip the patties; season with more salt and black pepper and top each with 1 cheese slice.
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continue cooking until the cheese melts around the edges and the burger is cooked through, about 1 more minute.
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spread the mayonnaise sauce on the bun tops.
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serve the patties on the buns with the tomatoes and lettuce.