Combine the basil, garlic, cheese, and vinegar in the bowl of a food processor. Pulse 3 or 4 times to mix. With the processor running, slowly add the olive oil through the feed tube until the pesto is well mixed. (It will be more liquid than paste-like). Season with salt.
Put the chicken in a glass or ceramic bowl and pour the pesto marinade over it, turning several times to coat. Cover and refrigerate for at least 3 hours and as long as 6 hours.
Prepare a charcoal or gas grill. Lightly spray the grill rack with vegetable oil cooking spray. The coals should be moderately hot to hot.
Lift the chicken from the marinade, letting most of the marinade drip back into the bowl. Grill the chicken for 12 to 16 minutes, turning several times and brushing with the marinade once or twice during the first 5 minutes of cooking, until cooked through.
Let the chicken cool. Remove the wing drumette, cut into bite-sized pieces, and chill for at least 1 hour.
Arrange the lettuce on a serving platter and sprinkle with lemon juice. Spoon the chicken over the lettuce, garnish with basil leaves, and serve.