Whisk milk, egg, and vanilla together in a liquid measuring cup and set aside.
In the bowl of a stand mixer fitted with a dough hook, combine flour, sugar, salt, and nutmeg. Whisk to combine, then sprinkle yeast on top.
With the mixer running on low, slowly pour in the warm milk mixture (should take about 30 seconds). Continue mixing on low 2-3 minutes, until the dough has mostly come together (there might still be some dry bits, that's okay). Increase the speed to medium and knead for 5 more minutes until the dough is soft and smooth.
Return the mixer to low speed and add the butter 1 tablespoon at a time, letting each piece fully incorporate before adding the next. This can take several minutes so, be patient and don't rush it. Once all of the butter has been added, increase the speed to medium and knead for another 5 minutes until the dough is very smooth, stretchy, and soft.
Shape the dough into a ball and place it in a lightly greased bowl to rise in the fridge for at least 3 hours, or as long as overnight (12-14 hours). (This is a good time to make your crumb topping!)
When the dough has doubled in size, turn it out onto a clean counter top. Deflate it gently with the pads of your fingers, then use a rolling pin to roll it out until it's ½" thick.
Use a big and a small round cutter to cut doughnut shapes out of the dough. Transfer doughnuts and doughnut holes to a lined sheet pan to rest. Use the smaller cutter to punch additional doughnut holes out of the scrap dough.
Cover the doughnuts on the sheet pan and let rise until slightly puffy. They should spring back with you poke them. This can take anywhere from 20 minutes to an hour depending on temperature, so be patient. You've got a pretty big window of time to get them fried once they're ready, and you can always pop them back in the fridge if they get too puffy.
Bring a pot of oil to 350°F. Arrange a cooling rack over a sheet pan and set aside.
Start by frying the doughnut holes as testers. 3-4 minutes total. Then fry the doughnuts, 3 minutes per side.
Transfer finished doughnuts to the cooling rack. They will be quite dark and the outsides will seem very crispy when they come out of the oil, but they'll soften up as they sit.
Let cool completely before glazing.