1. In the bowl of a food processor, pulse egg with salt and pepper until foamy and pale, about 1 1/2 minutes. With the machine running, add the oil through the feed tube, a drop at a time, until the mixture starts to thicken (do not stop the machine at this point or the mayonnaise may not come together). Add the remaining oil in a slow, steady stream.

2. When all the oil has been incorporated, slowly add lime juice, mixing until combined. Add basil; pulse until combined. Let chill before serving.