Heat the milk to 100 degrees. You can do this in the microwave or on the stove top.
Stir in the yeast until it is dissolved or close to dissolved.
Transfer to a medium-sized bowl and mix in 3/4 cup plus 2 tablespoons of the bread flour. Stir until smooth.
Cover the bowl (I like to use cloth bowl covers for this) to prevent drying out and place in a warm spot to rise.
When the batter has doubled in size (about an hour, depending on the temperature of your house), add it to a mixer fitted with a paddle attachment.
Add the egg yolks, superfine sugar, and salt. Mix on low speed until smooth.
Mix in vanilla extract and butter until smooth.
With the mixer running, slowly add the rest of the flour (1 1/4 cups + 2 tablespoons). Keep mixing until the batter just comes together into a smooth dough. It will be quite wet and sticky, but should be able to hold together as one “lump” of dough. If not, add a bit more flour, one tablespoon at a time.
Lift the ball of dough up and brush the bottom of the bowl lightly with vegetable oil.
Cover the bowl again and place in a warm spot to rise.
Once the dough has doubled in size (another hour or so), refrigerate it for 15 minutes. This will make it easier to roll out.
On a well-floured surface, gently roll out the dough to about 3/8” to 1/2” thickness.
Use cookie cutters to stamp out whatever shapes you like. I used a 3 1/2" circle for the large doughnuts with a 1 1/4" hole. For the minis, I used a 1 1/2" circle with a 15/16" hole (all of these cutters come in the same set). Save the holes to fry up as well, or roll the batter out again to make more doughnuts.
Place the stamped dough on a parchment- or Silpat-lined cookie sheet, cover with plastic, and allow to rise again in a warm spot.
Once the doughnuts are almost doubled in size (another hour or so), heat oil in a large pot or deep fryer until it reaches 360 F. I strongly prefer to fry in a deep fryer. It keeps everything so much cleaner and perfectly regulates the temperature.
Fry the doughnuts in batches until golden brown on each side (you'll need to flip them mid-fry). The total frying time will be under 2 minutes. Use tongs to flip and retrieve them.
Set hot doughnuts on paper towels to absorb extra oil.
Mix all ingredients in a small bowl until smooth.
Drizzle over piping hot doughnuts.
If necessary, reheat doughnuts in the microwave for about twenty seconds.