In a small saucepan over low heat, heat 3/4 cup/180 ml of the milk until it is warm to touch. Remove from heat and add one tablespoon of yeast. Stir until it dissolves and add 3/4 cup of flour. Stir until it is smooth. Cover with plastic wrap and leave to rest in a warm place for about 30 minutes.
In a large mixing bowl add 1 teaspoon of yeast with the remaining milk. Stir to combine. Add the rested flour mixture, the vanilla and egg yolks. Mix with a whisk until the ingredients are incorporated and the dough is smooth. This will only take less than a minute. Next, add 1 Cup of flour, sugar and salt. Mix now with a wooden spoon until the dough starts to come together. Add the softened butter in small pieces and mix until it is combined. You should have a smooth dough right about now.
With the remaining flour, start adding it in about 1/4 cup at a time. You are looking for a dough that is not sticky and pulls away from the sides of the pan. It should be quite soft. You may not need to use all of the flour. When I made them, I used all of the flour. When your dough has the right consistency, cover it and leave it in a warm place for about 30 minutes.
Once the time is up, punch the dough, cover and refrigerate it for at least an hour. At this stage, you can leave your dough in the fridge until later or even the next morning.
When you are ready to cut out your doughnuts, remove the dough from the fridge and transfer it to a lightly floured surface. Roll the dough out to about 1/2 inch thick. Using a 3 inch circle cutter of your choice, cut out your doughnuts. If you have a one inch circle cutter, cut out the middles. If not, improvise with whatever you have on hand. Place them on a floured baking tray and lightly cover with plastic wrap. Leave them to rise in a warm place. This will take about 30-40 minutes. After this time, they should have doubled in size. The dough will lightly spring back when touched when it is ready.
Heat your oil in a large deep sided skillet. Once the oil is up to temperature, start frying the doughnuts in batches. Cook them for about 1-2 minutes per side. When done, transfer them to a wire rack to drain and cool. Continue for remaining doughnuts.
Make the glaze by combining the powdered sugar, milk and vanilla in a bowl. Whisk to combine.
Dip in each doughnut face down (so to speak) and put it back on the rack to set. Ready to enjoy!