Prepare the dough by whisking the warm milk, yeast, and sugar together in a mixer. Cover and allow to sit for 5 minutes before adding in the eggs, butter, vanilla, nutmeg, salt, and 245g of flour.
Beat on low speed for 1 minute, scraping down the sides of the bowl as needed. Add remaining flour and beat on medium speed until the dough comes together - approximately 2 minutes. If needed, add more flour, a tablespoon at a time, until the dough pulls away from the sides of the bowl.
Knead the dough on a lightly floured surface for 2 minutes. Place the dough into a lightly oiled bowl, turning it to coat all sides in the oil. Cover and place the dough in a moderately warm environment for 1.5-2 hours.
When the dough is ready, punch it down to release the air and place it onto a lightly floured surface. Roll the dough out until it is 1/2 inch thick and cut it into 12 doughnuts using a doughnut cutter.
Line 2 baking sheets with parchment paper and place the doughnuts evenly on the tray. Loosely cover and allow to rest as you heat the oil.
Pour oil into a large pot set over medium heat. Heat the oil to 190°C, add 2-3 doughnuts at a time and cook for 1 minute on each side. Carefully remove with a metal spatula or slotted spoon.
Place the fried doughnuts onto a wire cooling rack.
Begin making the glaze by whisking all of the glaze ingredients together. Dip each warm doughnut (don’t wait for them to cool!) into the glaze, making sure to coat both sides.
Place back onto the prepared rack and allow to cool for 20 minutes until the glaze is set.