Combine the butter, basil, and salt and pepper to taste in a food processor and blend until smooth. Pack the mixture into a medium ramekin, cover with plastic wrap and refrigerate for at least 30 minutes. Split the lobsters in half lengthwise. Bring the knife down through the tail, making sure to cut just through the meat and to leave the shell connected. Lay the lobster open, leaving the two halves slightly attached. Pull off claws and legs from the lobsters and crack them slightly with the knife handle. Place the claws and legs on the grill over medium-low heat and cover with a pie pan. Cook them for 5 to 7 minutes per side. Brush the lobster bodies with the oil and season with salt and pepper to taste. Place them fleshside down on the grill over medium heat. Grill 8 to 10 minutes

Remove the lobsters and slather with the basil butter.