Preheat oven to 350 degrees. Line 2 muffin tins with paper liners.
In a large bowl combine cake mix, 3/4 cup eggnog, 1/2 cup vegetable oil, 1/2 cup unsweetened applesauce, 1 teaspoon vanilla extract, 1 teaspoon rum extract, and 1/4 teaspoon nutmeg, and mix on medium speed with a hand mixer until combined.
Add eggs one at a time, beating well after each addition.
Beat on high for 2-3 minutes until thick and creamy.
Fill each muffin cup with 1/2 cup of batter.
Bake in preheated oven for 15-20 minutes, until tops are springy and a toothpick inserted into the center comes out clean.
Cool in pan for 10 minutes and then remove to a wire rack to cool completely.
Frosting
Add 1 cup butter to the bowl of a stand mixer fitted with the paddle attachment. Mix on high until light and fluffy.
Add 1/2 teaspoon salt, 1 teaspoon vanilla, and 1/3 cup eggnog. Beat on medium high until well blended.
Reduce speed to to low. Add 1 cup confectioners sugar and mix until combined. Repeat three times with additional cups of confectioners sugar. Add more eggnog as needed to get to a smooth, piping consistency.
Transfer icing to a piping bag fitted with a large tip and pipe swirls onto cupcakes. Or just use a knife to spray a thick layer of frosting on each cupcake.
Eggnog cupcakes can be stored in the refrigerator for up to 5 days. You can also freeze unfrosted cupcakes for up to three months. Leftover icing can be stored in the fridge for two weeks. Bring to room temperature before using. You may even want to transfer to a bowl and whip with a hand mixer.