In a medium bowl, dissolve 2 tablespoons yeast in 3/4 cup warm milk. Stir in 3/4 cup of the flour to create a smooth paste. Cover and let rest in a warm spot for 30 minutes.
Combine the remaining warm milk and yeast in the bowl of a stand mixer fitted with the paddle attachment. Add the flour mixture along with the sugar, salt, vanilla, and egg yolks. Mix until smooth. Turn off the mixer and add 1/2 cup of the remaining flour. Mix on low for about 30 seconds. Add the butter and mix until it becomes incorporated, about 30 seconds. Switch to a dough hook and, with the mixer turned off, add more flour, about 1/4 cup at a time. Knead the dough on medium speed between additions until the dough pulls completely away from the sides of the bowl and is smooth and not too sticky. It will be very soft and moist, but not so sticky that you can’t roll it out. (You may have flour left over.) Cover the bowl with plastic wrap and refrigerate for at least 1 hour and up to 12 hours.
Line a baking sheet with a lightly floured dish towel. Roll out the dough on a lightly floured surface to 1/2 inch thick. With a doughnut or cookie cutter, cut out 3-inch-diameter rounds with 1-inch-diameter holes. (Note: If making filled doughnuts, clearly, don’t cut out the holes.) You can re-roll the scraps and cut out additional holes.
Place the doughnuts at least 1 inch apart on the baking sheet and cover loosely with plastic wrap. Let rest in a warm spot to proof until they almost double in size, 5 to 20 minutes, peeking every five minutes. To test whether the dough is ready, touch it lightly with a fingertip. If it springs back immediately, it needs more time. If it springs back slowly, it is ready. If it doesn’t spring back at all, it has over-proofed, in which case you can punch it down and re-roll it once.
While the doughnuts are proofing, heat a heavy-bottomed pot with at least 2 inches of oil until a deep-fat thermometer registers 360˚F (182°C). With a metal spatula, carefully place a couple of doughnut holes or doughnuts in the oil, being careful not to crowd the pot. Fry for 1 to 2 minutes per side, until light golden brown. Remove with a slotted spoon, drain on a wire rack over a paper towel, and let cool slightly before glazing. Repeat with the remaining doughnuts and holes, keeping the temperature consistent.