Halve the eggs lengthwise and place on platter or plate. Gently remove the yolks and transfer them to a food processor or bowl.
Add the mayonnaise and pesto and process or mash with a fork until smooth. Season with salt and pepper.
Transfer to the plastic storage bag and trim a corner. Pipe filling into egg white halves and garnish with additional salt, pepper, parmesan cheese, hot pepper flakes and fresh basil. Chill until ready to serve.
You could also drizzle a little of the pesto on top of the eggs just before serving.