Heat 2 quarts milk, the sugar, salt, vanilla seeds and pod, and cinnamon sticks in a large pot over medium-high heat, stirring until sugar dissolves and mixture is heated through. Remove from heat. Let stand for 30 minutes.
Prepare an ice-water bath. Whisk yolks in a medium bowl until pale, about 2 minutes. Whisk 1 cup of milk mixture into yolks in a slow, steady stream. Whisk yolk mixture into remaining milk mixture. Cook over medium-high heat, stirring constantly, until mixture registers 180 degrees on an instant-read thermometer, about 6 minutes. (Do not boil.)
Remove pot from heat, add melted bittersweet and milk chocolates, and stir until incorporated. Discard vanilla pod and cinnamon sticks. Pour mixture into a large bowl set in ice-water bath, and let cool, stirring often.
Whisk cream until soft peaks form. Pour cooled eggnog into a large serving bowl, and add brandy if using. (Add more milk to eggnog if necessary to reach desired consistency.) Top with whipped cream. Grate nutmeg over top, and sprinkle sparingly with cayenne. Serve immediately with churros on the side if desired.