Slice through the top 1/3 of each egg, ensuring some of the yolk is included in the slice. Carefully remove the egg yolk from the bottom portion of the egg using a spoon and transfer to a bowl.
Mix egg yolks, mayonnaise, mustard, pickle, horseradish, pickle juice, salt, and pepper together in a bowl. Fill a piping bag or a plastic bag with a corner snipped with the egg yolk mixture. Pipe egg yolk mixture into the hollowed egg whites creating the 'chick', ensuring there is enough filling to add 'eyes' and a 'beak.'
Slice small wedges out of the carrot rounds creating 12 'beaks'. Add a 'beak' to each 'chick'.
Slice olive pieces into small squares to be used as 'eyes.' Add 2 'eyes' to each 'chick.' Place the egg tops onto each 'chick' to look like a chick peaking out of the egg.