Prepare an ice-water bath; set aside. Whisk together eggs, egg yolks, sugar, salt, and the extracts in large saucepan over medium-low heat until combined. Pour in milk in a slow, steady stream, whisking until fully incorporated. Raise heat to medium. Cook, whisking constantly, until mixture registers 160 degrees on an instant-read thermometer and is thick enough to coat the back of a spoon, 30 to 35 minutes.
Pour milk mixture through a fine sieve into a medium bowl. Set bowl in ice-water bath; let mixture cool completely. Transfer custard to an airtight container, and refrigerate at least 4 hours or up to overnight.
Process candy canes in a food processor until finely crushed. Sift in a strainer to discard dust. Transfer to a shallow dish.
Melt chocolate in a heatproof bowl set over a pan of simmering water. Let cool 5 minutes. Dip rim of each cup into melted chocolate, then crushed peppermint. Refrigerate until set, about 5 minutes.
Remove custard from refrigerator; set aside. Put cream into the bowl of an electric mixer fitted with the whisk attachment; mix on medium speed until stiff peaks form. Fold into cold custard until combined. Fill cups with eggnog. Sprinkle with crushed peppermint, and garnish with peppermint sticks; serve.