First, prepare the dough. Whisk the warm milk, yeast, and sugar together. Set aside and allow to sit for 5 minutes. The mixture should be frothy on top after 5 minutes.
After 5 minutes, add the yeast mixture into a stand mixer. Add the eggs, butter, vanilla, nutmeg, salt, and 2 cups of the flour into the stand mixer. Beat on low speed and scrape down the sides of the bowl as needed. Add the remaining flour until the dough comes together, around 2 minutes.
Once the dough comes together, knead on a lightly floured surface and form into a smooth ball. Grease a large bowl with oil and place the dough in the bowl, ensuring it’s fully coated with oil on all sides. Cover and allow the rise for about 2 hour or until doubled in size.
Once the dough has risen, punch it down to release the air. Use a rolling pin to roll the dough out until it is 1/2 inch thick. Use a 3.5 inch donut cutter to cut into 12 donuts. Place the donuts on baking sheets lined with parchment paper and cover.
In a large pot, heat the oil to 375 F. Add 2-3 donut at a time and cook for a minute per side. Place the donuts on a wire rack. Now, make the glaze. Whisk all of the glaze ingredients together and dip the warm donuts into the glaze, making sure to coat both sets. Wait for the glaze to harden and then enjoy!