Start by warming the milk until it is just warm to the touch - it should not exceed 100 degrees F.
In the bowl of a stand mixer, add in the flour, yeast, and sugar. Then whisk it together by hand.
Add in the warmed milk, egg, butter, salt and vanilla extract. Knead the mixture with the dough hook on the stand mixer for 15-20 minutes on low speed until it is smooth and the bowl is clean.
Remove the dough and shape it into a ball. Lightly grease the mixing bowl that you were just using and place the ball of dough back in. Then cover the bowl with a towel or plastic and place somewhere warm for 30-45 min. The dough should double in size.
Once doubled in size, it is time to shape the donuts. *If you have an electric stove, start heating the oil now in a large heavy bottomed pot, as it will take longer.
Divide the dough into 6 equal weight portions.
Working with one portion at a time. First flatten slightly into a disk, then working in a clockwise direction, bring the edges into the center of the disk.
Shape your hand into a claw over the ball of dough with the smooth side face up. Roll the ball around in your hand on a clean work surface to create tension on the dough and seal the seem on the bottom of the ball.
Place the rolls onto a lightly floured lined baking sheet and cover with a sheet of plastic wrap. Place somewhere warm to rise.
If using a gas stove, heat the oil in a large heavy bottomed pot with a candy thermometer fitted to the side, over medium heat to 350 F.