Sprinkle yeast over the warmed milk and let sit until foamy, about 10 minutes. Once foamy, stir to dissolve.
In a stand mixer fitted with a paddle attachment, beat together ½ cup sugar and butter until fluffy, about 3 to 4 minutes.
Add yeast mixture, vanilla, salt, egg, and yolks and beat until combined.
With the motor running, slowly add flour and beat until all ingredients are incorporated. At this point, the dough will be somewhat sticky. If too sticky to handle, add more flour, 1 tbsp at a time, until it the dough just starts to pull away from the sides of the bowl with the mixer running.
Remove the dough and replace the paddle attachment with the hook attachment, return the dough to the bowl, and mix on medium for 4 to 5 minutes. The dough will still be somewhat sticky, but more elastic. Remove from bowl, and with slightly wet fingers, form the dough in a ball.
Transfer to a lightly greased bowl and cover loosely with plastic and set in a warm place until doubled in size, about 1½ to 2 hours.
On a lightly floured surface, roll dough into 1 14" round about ½" thick.
Using a 3" ring cutter, cut dough into rounds, gathering up scraps, and re-rolling to cut more rounds. You should end up with 16 to 20 rounds.
Transfer rounds to parchment paper-lined baking sheets, about 1" apart.
Cover loosely with plastic wrap and/or dry kitchen towel, and set in a warm place until doubled in size, about 30 more minutes.
Heat 2 to 3-inches oil in a 6-qt. Dutch oven until a deep-fry thermometer reads 350°F. Or, heat your deep fryer to 350°F.
Working in batches, carefully place the doughnuts in the hot oil and fry, flipping every 20 to 30 seconds, until puffed and golden, about 3 to 4 minutes.
Quickly add the fried doughnuts into a bowl with the remaining ½ cup of sugar and toss to coat. Place the doughnuts on a baking pan or platter.