Step 1In the bowl of a stand mixer, or in a large bowl, drizzle the yeast over the warm milk and stir to combine. Set the mixture aside for a few minutes to moisten the yeast.
Step 2Add the eggs and egg yolks to the milk mixture and gently beat to combine using the paddle attachment or a sturdy hand mixer. Add the sugar and vanilla and continue to beat until incorporated.
Step 3With the mixer running, add the flour, a spoonful at a time to incorporate, until three-fourths of the flour is added (if using a hand mixer, you may need to finish by hand as the dough may be too tough for the beaters). Add the salt and butter to the dough, mixing thoroughly to incorporate.
Step 4Add the remaining flour to the dough, gently but thoroughly beating to combine. The dough will be very soft and sticky, almost like a batter.
Step 5Grease a large bowl and place the dough inside. Loosely cover with plastic wrap and set aside in a warm place until doubled in volume, about 1 1/2 hours.
Step 6Turn the dough out onto a well-floured board and gently roll to a thickness of one-half inch. Using a doughnut cutter or 2 biscuit cutters (a larger one measuring 3 to 3 1/2 inches in diameter, and a smaller one measuring about 1 inch in diameter), cut the dough into doughnuts, spacing the doughnuts as close as possible. Collect the scraps, quickly knead to combine and roll the dough as before, cutting additional doughnuts.
Step 7Place the doughnuts on a floured sheet of parchment paper set on a baking sheet, leaving about 1 inch between each doughnut. Sprinkle a little flour over the tops of the doughnuts, then loosely cover with plastic wrap. Place the baking sheet in the refrigerator overnight.
Step 8The next morning, remove the baking sheet from the refrigerator and set the doughnuts aside for about an hour to warm slightly. Meanwhile, fill a deep fryer with oil, or fill a large pot to a depth of at least 3 inches, and heat to a temperature of 375 degrees.
Step 9Gently place the doughnuts in the fryer, being careful not to crowd. Fry the doughnuts on each side until puffed and golden, about 1 minute on each side.
Step 10Drain the doughnuts on a rack and cool slightly, then frost and decorate as desired.