01 - Place eggs in a saucepan and cover with cold water. Bring to a gentle boil over medium-high heat. Once boiling, cover, turn off the heat, and let eggs stand for 10 to 12 minutes.
02 - Transfer eggs to a bowl filled with ice water and cool for 5 minutes.
03 - Carefully peel the eggs, then cut each in half lengthwise.
04 - Gently remove the yolks and place them in a mixing bowl. Arrange the egg whites on a serving platter.
05 - Mash the yolks thoroughly with a fork. Stir in basil pesto, mayonnaise, 2 tablespoons grated Parmesan, lemon juice, black pepper, and salt. Mix until creamy and well combined.
06 - Using a spoon or piping bag, evenly fill each egg white half with the pesto-Parmesan mixture.
07 - Sprinkle with the remaining 1 tablespoon grated Parmesan and additional black pepper. Garnish with fresh basil leaves if desired.
08 - Serve immediately, or cover and refrigerate until ready to present.