Before starting, cut ~20 pieces of 6inX6in parchment squares out, as you will proof your doughnuts on separate pieces of parchment.
In a small bowl whisk together the warm water, yeast, and a pinch of sugar and let it rest until foamy (5 min).
In the bowl of a stand mixer, add flour, milk, remaining sugar, salt, eggs, vanilla, and yeast mixture. Mix these until combined.
Cover your bowl with a damp tea towel or plastic wrap, and let it rest at room temperature for 10 minutes.
Attach your dough hook to your stand mixer. Add room temperature butter to the dough, and knead your dough for 12-15 minutes on low. Cover your bowl with a damp tea towel or plastic wrap, and place it in the refrigerator for overnight for the best flavor, up to 12 hours. If you're in a hurry, and can compromise on flavor, you can also do your final rise now - keep it at room temperature 1-2 hours, until doubled in size.