In a small bowl, stir together the warm water, sugar, and active dry yeast. Let it sit for 10 minutes or until bubbly and frothy

In a separate large mixing bowl, whisk together the warm milk, melted butter, sugar, egg, vanilla, and salt. Whisk in the yeast mixture once frothy.

Add 2 ½ cups of flour and mix until a loose dough begins to form. Turn the mixture out onto the countertop and knead for 5 minutes, adding the remaining ¼ cup of flour as needed until a smooth and soft dough forms.

Shape into a ball and place the dough back into the bowl. Cover and let rise in a warm place for 1 hour or until doubled in size. Make the custard while the dough rises.

In a saucepan, whisk together the brown sugar and flour. Add about ½ cup of the eggnog and whisk until incorporated. (This prevents lumps of flour.) Add the remaining eggnog and whisk well. Bring the mixture to a boil over medium heat while stirring often. Boil for 2 minutes, stirring constantly until the custard thickens.

Immediately whisk in the butter, nutmeg, vanilla, and rum extract. Pour into a heat-safe bowl and cover with plastic wrap directly against the hot custard to prevent a skin from forming. Let cool down slightly before refrigerating for at least 1 hour or until fully chilled and set.

Roll out the dough on a floured countertop to a ½-inch thickness and cut out 12 donuts using a 2 ½-inch round cookie cutter. Arrange on a sheet pan that has been lined with parchment paper and dusted lightly with flour. Cover and let rise in a warm place for 30 minutes. While the donuts are rising, heat around 3-4 inches of oil in a heavy pot or Dutch oven over medium heat.

Once risen, poke a few small holes in each donut using a toothpick. This helps prevent large air pockets in the donuts. Fry the donuts in 350℉ oil until golden brown, flip, and continue frying the other side until golden brown. Transfer to a pan lined with paper towel to soak up the excess grease. Let the donuts cool completely before filling.

Use the handle of a wooden spoon to make a hole in the side of each donut and open up a cavity inside. Whisk together the melted butter, powdered sugar, nutmeg, rum extract, and enough eggnog* to make an icing that is thin enough to dip the tops of the donuts in. Transfer the chilled and set custard to a piping bag or Ziplock bag and pipe 2-3 tablespoons of custard into each donut. Dip each filled donut in the icing. Top with a sprinkle of nutmeg to garnish, if desired. Enjoy!