Boil eggs: Fill a large pot with water; bring to a boil over high. Carefully lower eggs into boiling water; cook, undisturbed, for 11 minutes, 30 seconds.
Make ice bath: Meanwhile, fill a large bowl halfway with ice; add water to cover. Set ice bath aside.
Chill eggs: Using a slotted spoon, immediately transfer cooked eggs from boiling water to ice bath. Let stand until completely cooled, at least 5 minutes.
Peel eggs: Working with 1 egg at a time, firmly tap on a flat surface until cracks form all over the shell. Peel under cold running water.
Halve eggs: Cut eggs in half lengthwise; remove yolks. Set egg white halves aside.
Make deviled egg filling: Using a fork, mask together yolks and mayonnaise in a medium bowl. Add relish, mustard, salt, and pepper; stir well to combine.
Fill eggs: Spoon or pipe about 1 tablespoon filling into each egg white half. Garnish with paprika, if desired. Serve immediately, or store, covered, in refrigerator up to 3 days.