In a small bowl, sprinkle the yeast over the milk. Stir briefly to combine and allow to sit until frothy, 5-10 minutes.
In a large bowl, whisk together the flour, sugar, salt, and nutmeg. Mix in the frothy yeast with milk, eggs, butter, and vanilla extract to form a dough.
On a lightly floured surface, knead the dough until smooth. If too sticky, add just enough flour to handle. It should still be a loose dough. Transfer to an oiled bowl, cover, and allow to rest at room temperature until doubled, 1-2 hours.
Gently deflate the risen dough, cover again, and refrigerate 8 hours to overnight.
Line two baking sheets with parchment. Remove the dough from the refrigerator. On a lightly floured surface, roll the dough into a 1/2 inch (1.25 cm) thick sheet. Use 2-3 1/2 inch (5-9 cm) star-shaped cookie cutters to cut out the doughnuts and arrange them on the prepared baking sheets about 1 1/2 inches (4 cm) apart. Cover and allow to rest until puffed, about 1 hour.
Fill a pan with vegetable oil at least 2 inches (5 cm) deep. Place over medium heat to 360˚F (182˚C). Line a baking sheet with paper towels.
Gently add the doughnuts, a few at a time to not overcrowd the pan, and cook until golden, about 1 minute per side. Adjust the heat between medium and medium low as needed to make sure the center cooks before the outside browns too much.
Remove to a towel-lined plate and repeat with remaining doughnuts. Allow to cool on a wire rack to room temperature before coating in glaze.