In a small bowl, combine the yeast, milk and 2 tablespoon of the sugar. Mix well, and leave to sit for 10-15 minutes, or until foamy.
In the bowl of a stand mixer fitted with the dough hook, combine the flour, salt, and remaining 2 tablespoon sugar. Mix briefly to combine. Add the eggs, vanilla, and foamy yeast mixture to the bowl. Mix on low for 2-3 minutes, until the dough is starting to come together. It may look slightly dry but do not worry - it will mix together nicely in the next steps. Increase the mixer speed to medium, and mix for another 10 minutes, until the dough is soft and smooth.
Reduce the mixer speed to low, and add the butter a little at a time, waiting until it is fully incorporated into the dough before adding the next piece. This process should take 3-4 minutes. Once the butter is fully incorporated, increase the mixer speed to medium, and mix for a further 5 minutes, until the dough is very soft and smooth.
Transfer to an oiled bowl, and cover tightly with plastic wrap. Place overnight in the fridge to rise.
Line two sheet pans with parchment paper squares. Turn the dough out onto a well floured surface. Roll until it is ½” (1.25cm) thickness.
Using a 1 ¾” (4.5cm) round cutter, cut out circles of dough. Using a very small round cutter or a milkshake straw, cut a circle out of the middle of each round of dough. Place onto the squares of parchment paper.
Lightly cover the cut-out doughnuts with plastic wrap, and leave to rise for 45-50 minutes, until puffy.
In the last 15-20 minutes of the doughnuts proofing, heat neutral oil to 350°f / 180°c in a large heavy-bottomed pot or a deep fryer.
Once the oil has come to temperature, gently lower the doughnuts, up to six at a time, into the hot oil. Cook, flipping every 30 seconds, for 3-4 minutes, until golden brown. Remove from the oil using a slotted spoon and place on a cooling rack. Allow to cool for 30 seconds before tossing in the sugar.
