Preheat oven to 350 degrees. Spray two donut pans with nonstick cooking spray. Set aside.
Cream together the butter and sugar in a mixing bowl. Add eggs and eggnog. Stir until well mixed.
In a separate bowl, combine the flour, baking powder, baking soda and salt. Stir gently to combine. With mixer on low, slowly add dry ingredients into the wet ingredients. Mix until just combined.
Fill the wells of the donut pan. Reduce oven temperature to 325 degrees and bake donuts for 8-10 minutes or until lightly golden. Let cool completely.
For the eggnog glaze, mix powdered sugar, eggnog, vanilla extract and nutmeg in a bowl. Stir to combine. Dip cooled donuts in glaze.
For the vanilla rum glaze, mix powdered sugar, nutmeg, vanilla extract and white rum in a bowl. Stir until smooth. Dip cooled donuts in glaze and let sit for 30 minutes+ for glaze to dry.