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1
To make the aioli, combine the mayonnaise, garlic, peppers, paprika, and salt in a food processor and process until smooth. Cover and refrigerate for at least 30 minutes before serving. The aioli can be prepared 1 day in advance, tightly covered, and refrigerated.
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2
Divide the meat into 4 equal portions (about 6 ounces each). Form each portion loosely into a 3/4-inch-thick burger and make a deep impression in the center with your thumb. Season both sides of each burger with salt and pepper. Cook the burgers, using 1 1/2 tablespoons of the oil.
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3
Heat a gas grill to high or heat coals in a charcoal grill until they glow bright orange and ash over. Brush the burgers with oil. Grill the burgers until golden brown and slightly charred on the first side, about 3 minutes. Flip over the burgers. Cook burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese; see step 3) or until cooked to desired degree of doneness.
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4
Heat a grill pan over high heat on top of the stove. Cook the burgers as for a grill, above.
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5
Heat the oil in the pan or griddle until the oil begins to shimmer. Cook the burgers until golden brown and slightly charred on the first side. Flip over the burgers. Cook burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese; see step 3) or until cooked to desired degree of doneness.
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6
Place the buns on a flat surface and spread half of the aioli over them. Top each one with a slice of the cheese, then a slice of the ham, then a burger, another slice of ham, and finally another slice of cheese, in that order. Spread the rest of the aioli on the bun tops and cover each burger with a top.
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7
Brush the tops of the buns with the remaining 1 1/2 tablespoons of oil and place on the grill, griddle, or saute pan, top side down. Using a heavy-duty metal spatula, press down on the bottom of the buns and grill until the tops are light golden brown, 1 to 2 minutes. Turn the burgers over and continue cooking, pressing down the tops, until the bottoms are light golden brown and the cheese has melted, about 1 minute longer. Serve immediately.
