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 crayfish 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 crayfish open, leaving the two halves slightly attached. Brush the lobster bodies with the oil and season with salt and pepper to taste. Place them fleshside down over medium coals and grill 8 to 10 minutes.
Remove the crayfish and slather with the basil butter.