Place cookies on a baking sheet. Microwave candy melts and oil in a glass bowl or measuring cup at 50% power for 1 minute. Stir. Continue melting at 50% power in 30-second increments (stirring after each) until smooth.
Dip a treat stick into melted candy, then insert into cookie filling, pressing down to sandwich. Refrigerate 5 minutes to firm.
Carefully cut jelly beans in half lengthwise, then cut each half in half crosswise (for a total of 4 pieces from each; these will become the beaks).
At any time, return melted candy to microwave for 15 seconds to keep fluid. Dip a cookie into melted candy, then shake off excess. Scrape back of cookie along bowl edge to keep from dripping. Add eyes and jelly bean beak. Hold cookie over a bowl and use a small spoon to add nonpareils at the bottom to resemble an eggshell. Secure stick in Styrofoam or place flat on a sheet of wax paper and continue with remaining chicks.