01 - Arrange eggs in a saucepan and cover with cold water. Bring to a gentle boil over medium-high heat, then cover, switch off the heat, and let stand for 10 to 12 minutes.
02 - Transfer cooked eggs to a bowl filled with ice water and allow to cool for 5 minutes.
03 - Carefully peel the cooled eggs and slice each one in half lengthwise.
04 - Gently remove yolks and place them in a mixing bowl, setting the egg whites on a serving platter. Mash yolks with a fork until smooth.
05 - Add pesto, mayonnaise, Parmesan cheese, lemon juice, black pepper, and salt to the yolks. Mix thoroughly until the mixture is smooth and creamy.
06 - Spoon or pipe the pesto-Parmesan mixture into each egg white half.
07 - Sprinkle tops with extra Parmesan and black pepper, and garnish with fresh basil leaves if desired. Serve immediately or refrigerate until ready to serve.