For the Doughnuts:

1. Activate the Yeast:
   • In a small bowl, combine the warm milk and granulated sugar. Stir until the sugar dissolves. Sprinkle the yeast over the milk and stir gently. Let it sit for 5-10 minutes, or until it becomes frothy (this means the yeast is active).
2. Mix the Dough:
   • In a large bowl, combine the yeast mixture, melted butter, salt, vanilla extract, nutmeg (if using), and eggs. Stir to combine.
   • Gradually add the flour, one cup at a time, mixing until a dough forms. You may need a little more flour depending on the humidity, but the dough should be soft and slightly sticky.
3. Knead the Dough:
   • Turn the dough out onto a floured surface and knead it for about 5-7 minutes until smooth and elastic. Alternatively, you can use a stand mixer with a dough hook on low speed to knead the dough for about 3-4 minutes.
4. First Rise:
   • Place the dough in a lightly greased bowl and cover it with a clean towel or plastic wrap. Let it rise in a warm place for about 1-1.5 hours, or until it doubles in size.
5. Shape the Doughnuts:
   • Once the dough has risen, punch it down to release the air. Turn the dough out onto a floured surface and roll it out to about 1/2 inch thick.
   • Use a doughnut cutter or two round cookie cutters (one large and one small) to cut out the doughnut shapes. Gather any leftover dough, re-roll it, and cut out more doughnuts.
6. Second Rise:
   • Place the doughnuts on a floured baking sheet, making sure they aren’t touching. Cover them with a towel and let them rise for another 30-45 minutes, or until puffy.
7. Fry the Doughnuts:
   • While the doughnuts are rising, heat about 2-3 inches of vegetable oil in a large pot or deep fryer to 350°F (175°C). You can test the oil’s temperature by dropping in a small piece of dough—it should sizzle immediately.
   • Carefully fry the doughnuts in batches for about 1-2 minutes per side, or until golden brown. Use tongs to flip them gently.
8. Drain the Doughnuts:
   • Remove the doughnuts from the oil and place them on a wire rack or paper towels to drain off excess oil.

For the Glaze:

1. Prepare the Glaze:
   • In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. If the glaze is too thick, add a little more milk; if it’s too thin, add more powdered sugar.
2. Glaze the Doughnuts:
   • While the doughnuts are still warm, dip them into the glaze, making sure they’re fully coated. Place them on a wire rack to let the glaze set.