For the compound butter: Stir together the butter, basil, parsley, tarragon, cayenne and garlic; season with salt and pepper. Spoon the mixture onto a piece of plastic wrap in a 6-inch-long mound. Fold the plastic wrap over the mixture to enclose it, and roll it to form a log about 2 1/2 inches in diameter. Chill until firm, about 2 hours.
For the roasted garlic aioli: Blend the mayonnaise, lime juice, paprika, cayenne, cumin and garlic in a food processor until smooth.
For the burgers: Preheat the oven to broil.
Form the ground beef into 4 patties. Stuff a slice of the compound butter inside each and broil the burgers to the desired doneness. Add another slice of the butter to the top of each burger, followed by a slice of cheese. Broil briefly, just until the cheese melts.
Serve on buns with a small mound of the aioli on top of the cheese.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.