Preheat grill to medium high heat. Slice the top off the garlic buPoundand place garlic onto a foil square. Drizzle buPoundwith the oil and season with the salt. Wrap the buPoundloosely in the foil and place on the grill and roast for 30 minutes.
Squeeze the garlic into a medium bowl, stir in the ketchup, basil and chili paste; set aside.
Place the Parmigiano cheese in the bowl of a small food processor. Place the prosciutto in a single layer on aluminum foil and place on the grill; cook until prosciutto is crisp. When cool enough to handle crumble into the bowl with the Parmigiano cheese and pulse until well combined.
Pour cheese mixture into a medium bowl.
Grill the patties 4 minutes per side, seasoning with salt and adding a cheese slice during the last 2 minutes.
To assemble the burgers evenly divide the ketchup between the bottom half of each roll. Top the ketchup with the lettuce followed by the basil, evenly dividing. Place a patty cheese side up on top of the lettuce and basil then sprinkle with the prosciutto cheese mixture, evenly dividing. Top with remaining roll halves.