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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.
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2
Cover and refrigerate for at least 30 minutes before serving.
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3
The aioli can be prepared 1 day in advance, tightly covered, and refrigerated.
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4
Divide the meat into 4 equal portions (about 6 ounces each).
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5
Form each portion loosely into a 3/4-inch-thick burger and make a deep depression in the center with your thumb.
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6
Season both sides of each burger with salt and pepper.
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7
Cook the burgers, using 1 1/2 tablespoons of the oil (see page 16).
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8
Place the bun bottoms on a flat surface and spread half of the aioli over them.
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9
Top each one with a slice of cheese, then a slice of the ham, then a burger, another slice of ham, and finally another slice of cheese, in that order.
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10
Spread the rest of the aioli on the bun tops and cover each burger with a top.
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11
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.
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12
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.
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13
Turn the burgers over and continue cooking, pressing down on the tops, until the bottoms are light golden brown and the cheese has melted, about 1 minute longer.
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14
Serve immediately.
