1. Place eggs in a large saucepan and cover with cold water by 1 inch. Cover and bring to a boil over high heat. Remove from heat and let stand, covered, for 12 minutes.
2. Fill a large bowl with ice water. Transfer eggs to bowl; let cool completely, about 10 minutes. Tap each egg on the counter to crack shell; return to ice water. Peel eggs under cold running water, 1 at a time. Cut eggs in half lengthwise and, using a teaspoon, remove yolks to a small bowl. Cover and refrigerate whites.
3. Bring a small pot of water to a boil. Add basil, return to a boil and cook until bright green, about 1 minute. Drain, rinse with cool water, drain again and squeeze out excess moisture. Transfer to a food processor. Add yolks, mayonnaise, salt and pepper. Blend until smooth. Taste and adjust seasoning.
4. Fill each egg-white cavity with about 1 heaping tsp. yolk mixture and place on a platter. Serve or tent with foil and chill until ready to serve, up to 6 hours.