Cook Eggs Place eggs in a single layer in a medium saucepan. Add enough cold water to cover the eggs by 1 inch. Bring to a full rolling boil over high heat. Remove from heat, cover and let stand 15 minutes; drain. Place eggs in ice water until cool enough to handle; drain.
Peel Eggs To peel eggs, gently tap each egg on the countertop. Roll egg on countertop under the palm of your hand. Peel off eggshell, starting at the large end.
Grate Eggs Using the large holes of a box grater, shred eggs into a medium bowl. Use a fork to coarsely mash the eggs.
Finish and Serve Stir in relish, shallot, and capers. Add Greek yogurt, mayonnaise, mustard, vinegar, hot pepper sauce, salt, and pepper. Fold in until combined. Fold in 1 Tbsp. chives. Top with remaining 1 Tbsp. chives and paprika. Serve with vegetable crudités and/or crackers.
