Cook 1/2 cup sugar in a large saucepan over medium-high heat, lifting and tilting pan often, until sugar melts and turns deep amber, 6 to 8 minutes. Remove from heat; at once, add 3 cups milk, cinnamon sticks, and vanilla bean.
Return pan to medium heat and stir until caramelized sugar dissolves, 5 to 8 minutes. Let cool; cover and chill at least 3 hours or, preferably, overnight.
Meanwhile, in a large bowl, whisk eggs to blend well. Fill a large bowl with ice and water; set aside. In a medium saucepan over medium-high heat, heat remaining 3 cups milk and 3/4 cup sugar to steaming, stirring often. Whisk into eggs, then return to saucepan. Reduce heat to medium-low. Cook, stirring, until custard thickly coats a metal spoon and reaches 160° on an instant-read thermometer, 15 to 25 minutes. Set pan in ice bath and stir occasionally until cold, about 10 minutes. Chill in refrigerator if making ahead.
Discard cinnamon sticks and vanilla bean from caramelized sugar mixture and pour into a 4- to 5-qt. punch bowl. Stir in custard.
In mixer bowl, whip cream until it holds soft peaks. Whisk half of cream into eggnog and spoon remaining cream on top. Dust generously with nutmeg. Dip spoon down into eggnog as you serve, and add rye or bourbon to glasses if you like.
MAKE AHEAD Through step 4, chilled, up to 2 days; whisk to blend.