In the bowl of a stand mixer, mix together the warm milk, yeast, and one tablespoon of the sugar. Allow the mixture to sit for 5-10 minutes until it is foamy.
Add the flour, eggs, vanilla, and the remaining sugar to the yeast mixture. Using the dough hook attachment, knead the dough on medium-low speed until it is just combined.
Add the softened butter and increase the speed to medium. Knead the dough for 8-10 minutes, or until it is smooth and elastic.
Cover the bowl and let the dough rise for one hour in a warm spot, or until it has doubled in size.
Once the dough has risen, punch it down and transfer it to a lightly floured surface. Roll the dough into a 10x13 inch rectangle about 1/2 inch in thickness.
Use a biscuit cutter or drinking glass about 3 inches in diameter to cut out 12 donuts. If desired, re-roll the scraps and cut out as many donuts as you can. Lightly cover the donuts with greased plastic wrap and allow them to rise for an hour.
Heat about three inches of oil in a large pot. Once the oil has reached 350°F, add a few donuts at a time (do not overcrowd the pot). Fry the donuts on each side for a few minutes, or until they are golden brown.
Remove the donuts with a slotted spoon and transfer them to a paper towel-lined plate to absorb any excess oil, then toss them in a bowl with sugar to coat all sides. Repeat with the remaining donuts.
Allow the donuts to fully cool. Use a butter knife to make a small hole in the side of each donut. Gently squeeze your favorite filling into the donuts with a piping bag. Serve immediately and enjoy!