16 large head-on jumbo shrimp in the shell, shells split down the back
Put a large grill pan on 2 burners over medium-high heat or preheat an outdoor gas or charcoal grill and get it very hot. If you're using an outdoor grill, take a few paper towels and fold them over several times to make a thick square. Blot a small amount of oil on the paper towel, then carefully and quickly wipe the hot grates of the grill to make a nonstick grilling surface.
Meanwhile, add the butter into a food processor with the basil leaves (reserve a few for garnish), the juice of 1 of the lemons, and salt and pepper. Puree. Stuff about half of the butter under the shells of the shrimp (about 1/2 tablespoon per shrimp). Lay the shrimp on the hot grill and cook for 3 minutes on each side, brushing with the remaining basil butter a few times as they cook.
To serve, put the shrimp on plates and dot with the remaining basil butter. Squeeze the remaining lemon over the shrimp and garnish the plates with basil leaves.