01 - Arrange eggs in a saucepan and cover completely with cold water. Heat over medium-high to a gentle boil, cover, remove from the heat, and allow eggs to rest for 10 to 12 minutes.
02 - Using tongs, transfer eggs to a bowl of ice water and cool for 5 minutes to halt cooking.
03 - Carefully peel the cooled eggs and slice each in half lengthwise with a sharp knife.
04 - Remove yolks and place in a mixing bowl, arranging egg white halves on a serving platter. Mash yolks thoroughly with a fork.
05 - Add basil pesto, mayonnaise, Parmesan cheese, lemon juice, black pepper, and salt to the mashed yolks. Mix until smooth and creamy.
06 - Spoon or pipe the pesto-Parmesan filling into the hollow centers of each egg white half.
07 - Dust filled eggs with extra Parmesan and cracked black pepper; top with fresh basil leaves if desired. Serve immediately or refrigerate until service.