You can make the meringue a week ahead and store in an airtight container in a cool, dry place (though not in the freezer or refrigerator). Any humidity or moisture will soften the meringue.
3 large egg whites
3/4 cup sugar
1/2 teaspoon pure peppermint extract
Red gel-paste food coloring
1 cup heavy cream
6 ounces good-quality semisweet chocolate, finely chopped
Preheat oven to 175 degrees. Line 2 baking sheets with parchment paper; secure corners with masking tape. Fit a pastry bag with a small open-star tip (such as Ateco #22). Set aside.
Make meringues: Put egg whites and sugar and cream of tartar in the heatproof bowl of an electric mixer. Set bowl over a pan of simmering water, and stir gently until sugar has dissolved and mixture is warm to the touch, 2 to 3 minutes.
Transfer bowl to an electric mixer fitted with the whisk attachment. Mix on medium-high speed until stiff peaks form. Mix in peppermint extract.
Using a new small paintbrush, paint 2 or 3 stripes of red food coloring inside the pastry bag. Fill bag with 1 to 2 cups meringue. Pipe small (3/4-inch-high) star shapes onto prepared baking sheets. Refill bag as necessary, adding food coloring each time.
Bake cookies until crisp but not brown, about 1 hour 40 minutes. Let cool completely on sheets on wire racks.
Meanwhile, make ganache: Heat cream in a small saucepan over medium-low heat until just simmering. Pour over chocolate in a small bowl. Let stand 5 minutes. Gently stir until smooth, about 5 minutes. Let ganache cool at room temperature, stirring every 5 to 10 minutes, until thick enough to hold its shape, about 45 minutes. (If ganache sets before using, reheat in a heatproof bowl set over a pan of simmering water; repeat the cooling process.)
Before serving or mailing, fill a pastry bag fitted with a small plain round tip with ganache. Pipe a small amount onto bottom of 1 meringue. Sandwich with another. Repeat with remaining ganache and meringues. Transfer to wire racks; let set 30 minutes.