Combine 2 tbsp yeast, ¾ cup milk and ½ cup flour in a medium bowl and let rest for 30 minutes in a warm place.
In the bowl of a stand mixer, with the paddle attachment fitted, combine the rest of the yeast and milk with the egg yolks, vanilla, salt and sugar. Add in the butter and ½ cup of flour - mix on low until incorporated.
Change to the dough hook, then knead in the rest of the flour - ¼ cup at a time. Continue to knead until the dough is smooth and elastic. Transfer to an oiled bowl, cover and rest for 1 hour.
Roll the dough out on a floured surface until ½ inch thick. Cut out circles using a 3" cutter, and then a 1" cutter (see notes) to make the hole. Transfer the doughnuts (and doughnut holes) to a cookie sheet lined with a floured kitchen towel - spaced at least 1" apart. Re-roll the scraps and repeat the cutting + transferring.
Cover loosely with cling film and rest for 5-20 minutes (see notes).
Heat a 2 inch depth of oil to 360 F (180 C) in a deep, heavy bottomed pan. Use a metal spatula to transfer 2 or 3 proved doughnuts to the frying pan. Fry for 3-4 minutes until golden brown, flip and fry for a further 3-4 minutes. Remove from the oil using a slotted spoon and drain on paper towel. Repeat with the remaining doughnuts and doughnut holes. Let cool slightly then glaze.