Preheat the oven to 350º F. Lightly grease two 6-count donut pans with a bit of butter or extra virgin olive oil and set aside.
Whisk together the flour, sugar, baking powder, baking soda, salt, and nutmeg in a large mixing bowl.
With an electric hand mixer or stand mixer (with the paddle attachment), slowly and gently beat in the eggnog, egg, extra virgin olive oil, and vanilla extract. Mix until all of the ingredients are incorporated.
Spoon the mixture into a large piping bag or ziplock bag and cut a 1/2-inch hole at the tip.
Carefully pipe the mixture about 3/4 of the way into each donut mold.
Place in the oven and bake for about 8-10 minutes, or until the donuts are slightly firm and a cake tester comes out clean.
Remove the donuts from the oven and set them aside to cool completely before adding glaze.
To make the glaze, whisk the powdered sugar, eggnog, and cinnamon together in a medium-sized bowl. Taste, and adjust as you see fit.
Once the donuts cool, you can either dunk the tops of each donut into the glaze for full coverage, or simply drizzle the glaze over the donuts, as pictured in the recipe post.
Store in an airtight container for up to 3-4 days.