Slice the eggs in half and put the whites on a platter. Put the yolks into the bowl of a food processor.
Add the avocado, lemon juice, mustard, pesto, and pepper flakes into the food processor along with a few pinches of salt. Process until smooth.
Put the mixture into a piping bag and pipe it into the egg whites.
Top each egg with a few pieces of julienned pickled carrot and a little basil. Enjoy!