Peel hard-boiled eggs and cut them lengthwise. Carefully remove yolks; place 4 yolks in each of three bowls. Set whites aside. Mash yolks with a fork.
Add mayonnaise and curry powder to one bowl; mix. Season with salt and pepper.
Place basil and mayonnaise in bowl of a food processor; blend about 2 minutes. Add basil mixture to second bowl of yolks; mix. Season with salt and pepper.
Add creme fraiche, mustard, and cornichons to last bowl of yolks; mix. Season with salt and pepper.
Fill eight reserved egg-white halves with each filling; serve.
Prepare an ice-water bath. Place eggs in a medium saucepan; add enough water to cover by 1 inch. Bring to a boil over medium-high heat. Cook, stirring gently, as water begins to boil, 2 minutes. Cover and remove from heat. Let stand 10 minutes. Transfer eggs to ice-water bath to cool.
Peel eggs and halve lengthwise and/or crosswise. Remove yolks and transfer to the bowl of a food processor along with mayonnaise, lemon juice, mustard, and cayenne pepper; season with salt and pepper and process until smooth.
Transfer yolk mixture to a pastry bag fitted with an open-star tip (such as Ateco no. 825). Pipe mixture into egg white halves, filling to about 1/2 inch over surface.
Peel eggs and cut in half, either lengthwise or crosswise. Carefully scoop out yolks, and arrange whites on a platter or tray.
In the bowl of a food processor, process egg yolks and mayonnaise until smooth. Alternatively, press yolks through a sieve; mix yolks with mayonnaise until smooth. Season with salt, black pepper, and cayenne.
Transfer egg yolk mixture to a pastry bag fitted with a large plain or star tip. Pipe mixture carefully and very neatly into egg halves; top with assorted garnishes, if desired. Cover with plastic wrap, and refrigerate until ready to use, up to 3 hours.