Turn a gas grill to medium high and place the pepper directly over the flame. Cook, turning it with tongs as the underside blisters and chars, until the whole pepper is blistered. Place in a bowl and cover with a dishtowel until cooked. Peel off the skin and discard any seeds and rubs from the inside of the pepper. Coarsely chop the pepper. (You can also do this underneath the broiler, on a rimmed baking sheet).

Combine pepper and garlic in a food processor and pulse to combine. Pulse in the basil and mayonnaise, then let the motor run until all is smooth and incorporated. Season with salt and pepper.