Fill a pot with water and add the eggs. Bring to a full boil on the stove top and cook for 10 minutes. Immediately transfer to an ice bath and allow eggs to chill at least 10 minutes.
While you’re cooking or chilling your eggs, add the avocado, basil, pesto sauce, mustard, and sea salt to a food processor.
Peel the eggs. Slice them in half length-wise and scoop out the cooked yolks. Add the yolks to the food processor with the rest of the ingredients.
Process this mixture until it’s very creamy. You may need to stop your food processor a couple of times to scrape the sides with a rubber spatula to help it move along.
Fill the egg whites with the avocado pesto mixture, garnish with chives, and enjoy!