Make the dough: Add the unsalted butter to a heat-safe bowl or liquid measuring cup and microwave for about 30 seconds. It should be soft and starting to melt.
Add the whole milk and microwave in 15-second intervals until the temperature reaches 100-110°F on an instant read thermometer (about 45-60 seconds more).
Mix in the granulated sugar and active-dry yeast. Let that sit for 5-10 minutes. The yeast should dissolve and there should be bubbles forming on the surface.
Transfer the mixture to a large bowl or the bowl of a stand mixer fitted with a dough hook attachment. Mix in the eggs and vanilla extract.
Add half of the flour, salt, and ground nutmeg and mix until combined. Add the rest of the flour and mix until the dough comes together. Knead the dough by hand or in the stand mixer for about 5 minutes. The dough will be very soft and slightly sticky.
First rise: Stretch the dough to form into a smooth ball and place it in a large bowl that has been lightly greased with butter. Cover the bowl with plastic wrap and let the dough rise for 90 minutes or until doubled in size*. The dough will rise faster in a warm area, such as next to a window on a sunny day or near a warm oven. You can also let the dough rise overnight in the fridge. Take it out of the fridge and let the dough come to room temperature before continuing with the recipe.
Shape the dough: Punch the dough down to release the air. Divide it into 15 equal pieces and roll each into a round ball.
Use your hands to gently flatten each ball into a half-inch thick circle. Arrange the shaped dough on a baking mat or a baking sheet lined with parchment paper. Cover loosely with plastic wrap and let rise until puffed up slightly. About 60-90 minutes. Again, they will rise faster in a warm area, or you can let them rise overnight in the fridge. Take them out of the fridge and let the dough come to room temperature the next day before frying.
Fill a large, heavy-bottom pot with oil, about 1-2 inches from the top of the pot. Heat the oil to 350°F. Use a deep-frying or candy thermometer for the best results. If the oil temperature is too low the donuts will absorb more oil and come out greasy. You also don't want the oil too hot or the donuts will burn on the outside before fully cooking in the center.
Working in batches of 2 or 3 at a time, fry the donuts for 5 minutes, flipping halfway. Remove from the oil and set on a wire rack to drain any excess oil.
Toss the donuts in a bowl of granulated sugar to coat.