Make the aioli: in a food processor—combine mayo, garlic, peppers, paprika, and salt; process until smooth.
Cover and chill in refrigerator for at least 30 minutes before serving (can be prepared 1 day in advance, tightly covered, and refrigerated).
Make the burgers: divide the meat into 4 equal portions (~6 oz each); form each portion loosely into a ¾ inch thick patty and make a deep depression in the center with your thumb.
Season both sides of each patty with salt and pepper; using 1 ½ tablespoons oil, brush the patties with the oil.
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.
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).
Place the bun bottoms on a flat surface and spread half of the aioli over them.
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.
Spread the rest of the aioli on the top buns and cover each burger with a top.
Brush the tops of the buns with the remaining 1 ½ tablespoons oil and place on the grill 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, 1-2 minutes.
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.