Hard boil the eggs and let cool quickly in ice water. Change out the ice water when it gets warm. If you cool the eggs slowly, a greenish skin forms around the yolks inside.
Peel the eggs and rinse. Pat dry.
With a sharp paring knife, cut about 25% off the smaller end of each egg, revealing the cooked yolk.
Carefully remove the yolks from the whites and put them in a separate bowl.
Rinse and pat dry the whites. With your paring knife, cut around the edge of the small end of each egg in a zigzag pattern. Cut a small slice off the other end of each egg to make them flat so they will stand up on their own.
To the egg yolks, add the mustard, mayo, relish, sugar, paprika, cayenne, salt and ground pepper. Mash the yolks with a fork and mix all ingredients well.
Spoon 1/6 of the egg yolk mixture into each of the larger, upright white parts. Cap each with a jagged small piece, leaving part of the yolk mixture visible. Push two whole peppercorns into each visible yolk part for eyes and a small piece of carrot into each for a beak.