A great mashup of holiday traditions. And who doesn't love a little booze in their baked goods. If nothing else it should make it easier to handle all that time with relatives.
Add the yeast to the water, stir, and allow to bloom for five minutes.
Add the eggnog and the butter to a small saucepan and place over low heat until the butter is melted. Whisk to combine and allow to cool to room temperature.
Add the yeast mixture and the eggnog and butter mixture into the bowl of a stand mixer.
Add the eggs, salt, and half of the flour. Using the paddle attachment, combine the ingredients on low speed until flour is incorporated and then turn the speed up to medium and beat until well combined.
Add the remaining flour, combining on low speed at first, and then increase the speed to medium. Once all the flour is incorporated switch to the dough hook attachment of the mixer and knead on medium speed until the dough pulls away from the bowl and becomes smooth.
Spray a large bowl with cooking spray and and transfer the dough to it. Cover with plastic wrap and let rise until doubled in size, about one hour.
On a well-floured surface, roll out dough to 1/2 inch thick. Cut out dough using a a large cookie cutter or pastry ring. Let the cut doughnuts rise for thirty minutes on floured baking sheet, covered lightly with a tea towel.
Preheat the oil in a large pot or Dutch oven to 350°F. Slowly and carefully, place the doughnuts into the oil, being careful not to splash. Work in batches so the doughnuts have plenty of room to cook and the oil temperature does not drop too much.
Cook on each side until golden brown. Transfer to a cooling rack placed in a baking pan. Allow to cool for a half an hour prior to filling.
In a large bowl, whisk together the pudding mix, bourbon, rum, and eggnog. Allow to set in the fridge until thickened.
Add the filling to a piping bag with a filling tip and twist closed. From the top of doughnut, press down with the filling tip about 3/4 of the way and wiggle it around to create a pocket. Be careful to make sure you do not make the hole on the top too big. Squeeze the piping bag lightly until the filling just starts to come of the hole. Eat immediately!