Prepare perfect hard-boiled eggs. Place the eggs in a medium saucepan and cover with cold water by about an inch. Bring to a rolling boil over high heat, then immediately remove from heat, cover, and let stand for exactly 10 minutes.
Cool them down. Transfer the eggs to an ice water bath and let them chill for at least 5 minutes. This makes them easier to peel and stops that unappealing gray ring from forming around the yolk.
Peel with care. Gently tap each egg on the counter and roll it to create cracks all over, then peel under a thin stream of cold water to help remove the shell cleanly.
Make strategic cuts. Pat the eggs dry with paper towels. Using a sharp knife, cut each egg in half lengthwise. For standard deviled eggs, you'd cut directly through the middle, but for our chicks, cut about 1/3 from the bottom and 2/3 from the top to create the chick's "body" (larger bottom piece) and "head" (smaller top piece).
Extract the yolks. Carefully remove the yolks from both halves and place them in a small mixing bowl. Set the egg white pieces on your work surface, with the larger bottom pieces forming the bodies and the smaller top pieces forming the heads.
Make it smooth and creamy. Mash the yolks with a fork until no large pieces remain. Add the mayonnaise, Dijon mustard, vinegar, salt, pepper, and a pinch of paprika. Mix until completely smooth and creamy. If the mixture seems too stiff, add a bit more mayonnaise, 1/2 teaspoon at a time.
Fill those chicks. Spoon the yolk mixture into the larger bottom pieces of the egg whites (the "bodies"), mounding it slightly higher than the edge. Alternatively, use a piping bag with a round tip for a more polished look.
Assemble your chicks. Place the smaller egg white pieces (the "heads") at an angle partially on top of the yolk-filled bottom pieces, pressing gently to secure them.
Add personality. Insert small carrot triangles between the head and body to create beaks. Add tiny olive or caper pieces for eyes.
Final touches. If desired, place your chicks on a bed of fresh dill or parsley "nests" on your serving platter. For extra Easter charm, add a few black peppercorns beside some of the chicks to look like little eggs.
Serve with pride. Refrigerate until ready to serve, ideally within a few hours of assembly for the freshest presentation.