Cut eggs in half lengthwise. Remove yolks. Process yolks, mayonnaise, and next 4 ingredients in a food processor until smooth, stopping to scrape down sides.
Spoon filling into egg whites. Cover and chill up to 8 hours. Garnish, if desired.
Bacon-Stuffed Eggs: Stir 2/3 cup (8 slices) crumbled cooked bacon, 3 tablespoons pickle relish, and 1/4 cup chopped fresh chives into egg mixture. Proceed as directed.
Shrimp-Stuffed Eggs: Stir 3/4 pound shrimp, cooked and chopped; 2 tablespoons prepared horseradish; and 6 green onions, minced, into egg mixture. Proceed as directed.