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1
Make the aioli: in a food processor-combine mayo, garlic, peppers, paprika, and salt; process until smooth.
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2
Cover and chill in refrigerator for at least 30 minutes before serving (can be prepared 1 day in advance, tightly covered, and refrigerated).
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3
Make the burgers: divide the meat into 4 equal portions (~6 oz each); form each portion loosely into a 3/4 inch thick patty and make a deep depression in the center with your thumb.
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4
Season both sides of each patty with salt and pepper; using 1 1/2 tablespoons oil, brush the patties with the oil.
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5
Heat a grill pan over high heat on top of stove; cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef, 5 minutes for turkey.
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6
Flip burgers over; cook beef burgers until golden brown and slightly charred on second side, 4 minutes for medium rare or until cooked to desired doneness (cook turkey burgers until cooked throughout, about 5 minutes on second side).
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7
Place the bun bottoms on a flat surface and spread half of the aioli over them.
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8
Top each one with a slice of cheese, then a slice of ham, then a burger, another slice of ham, and finally another slice of cheese, in that order.
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9
Spread the rest of the aioli on the top buns and cover each burger with a top.
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10
Brush the tops of the buns with the remaining 1 1/2 tablespoons oil and place on the grill pan, top side down.
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11
Using a heavy-duty metal spatula, press down on the bottom of the buns and grill until the tops are light golden brown, 1-2 minutes.
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12
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.
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13
Serve immediately.
