In a food processor, pulse the egg with the salt and pepper until foamy and pale, about 1 1/2 minutes.
With the machine running, add the oil through the feed tube, 1 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.
When all the oil has been incorporated, slowly add the lime juice, mixing until combined.
Add the basil; pulse until combined.
Let chill before serving.
Fresh mayonnaise can be refrigerated in an airtight container up to 5 days.