Preheat the oven to 350 degrees. Line cupcake tin with paper lines. In the bowl of your stand mixer add the eggnog, melted butter, vinegar, vanilla, and sugar. Beat on medium speed until well combined. Sprinkle the flour on top of the eggnog mixture and then sprinkle the baking soda, baking powder, salt, and nutmeg on top of the flour. Mix on low speed just until everything is mixed in.
Fill the lined cups up 2/3 full with the prepared batter. Bake for 22-24 minutes, or until a toothpick inserted in the middle comes out clean. Let the cupcakes cool in the pan for a few minutes before you transfer them to a wire rack. Let them cool completely before decorating.
For the icing:
In the bowl of your stand mixer with the whisk attachment, beat the butter, eggnog, salt, vanilla, nutmeg, and 1 cup of powdered sugar on medium-high speed until the mixture is very light and fluffy, around 3 minutes. Add the remaining powdered sugar, one cup at a time until you get a medium consistency icing. Scrape the edges of the bowl down as you go.
Put the icing in a pastry bag and pipe onto cool cupcakes. Garnish with a sprinkle of cinnamon or a cinnamon stick if you like.
Notes: You don't have to use a stand mixer for the cupcake batter, but using one for the icing really lends to a light and fluffy icing.