To give these cups their festive design, paint lines of red gel-paste food coloring up the inside of an empty pastry bag before adding the meringue.
Preheat oven to 175 degrees. Trace 16 circles onto each of 2 pieces of parchment using a 1 3/4-inch round cookie cutter, spacing about 2 inches apart. Coat 2 baking sheets with cooking spray, and top each with a sheet of parchment, traced side down.
Make the meringue cups: Heat egg whites and sugar in a heatproof bowl of a mixer set over a pan of simmering water, stirring, until sugar dissolves and mixture is warm to the touch, about 3 minutes. Transfer bowl to mixer. Whisk on medium-high speed until stiff peaks form. Beat in peppermint extract.
Using a small food-safe paintbrush, paint 3 stripes of red food coloring up the inside of a pastry bag fitted with a 1/4-inch plain round tip. Transfer egg-white mixture to pastry bag; pipe coils to fill each traced circle. Pipe up sides of each to create 3/4- to 1-inch-high cups.
Bake until meringue cups are crisp but not browned and lift off parchment easily, about 1 hour 30 minutes. Transfer meringue cups to a wire rack, and let cool completely.
Meanwhile, make the ganache: Transfer chocolate to a heatproof bowl. Bring cream to a gentle simmer in a small saucepan over medium heat. Pour over chocolate, and let stand for 2 minutes. Stir until smooth. Cover loosely with plastic wrap, and cut a small slit in the top. Refrigerate until firm, 1 hour to 1 hour 15 minutes.
Stir ganache, and transfer to a pastry bag fitted with a 1/4-inch open-star tip. Pipe ganache into the center of each meringue cup. Garnish with candy cane.