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1
Place feed shaft, blade, and 1/4-inch die of meat grinder in freezer until well-chilled. Meanwhile, place meat chunks on rimmed baking sheet, leaving space between each piece and place in freezer for 10 minutes until meat is firm, but not frozen.
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2
Combine meat in large bowl and toss to combine. Pass meat through grinder twice. Form into four disks, about 2-inches tall, and 2.5-inches wide. Refrigerate until ready for use.
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3
Open buns but do not split hinge. Brush lightly with butter, then place under broiler or in toaster oven until golden brown, about 1 minute. Spread 1 tablespoon Shack Sauce on top half of each bun (for true authenticity, squirt out of squeeze bottle into three lines, three passes on each line). Place 1 leaf lettuce and 2 slices tomato on top half of each bun.
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4
Using wadded-up paper towel, rub inside of heavy-bottomed 12-inch skillet with vegetable oil, then place over medium-high heat until just beginning to smoke. Season beef pucks on top side with salt and pepper, then place, seasoned side down, in skillet. Using back of heavy, flat spatula, press down on beef pucks firmly to form 4-inch round patties, being careful not to let it stick to bottom of spatula. Season top side with salt and pepper. Cook until crisp brown crust has formed, about 2-minutes.
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5
Carefully scrape patties from skillet, and flip. Top each patty with 1 slice American cheese. Cook until cheese is melted, about 1 minute longer. Transfer patties to burger bun bottoms, close sandwiches, and serve.
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6
Combine all ingredients in blender until smooth, scraping down sides of blender with rubber spatula as necessary. Makes about 3/4 cup sauce.
