Make the Doughnuts:

1. Activate the yeast: In a small bowl, combine the warm milk, sugar, and yeast. Stir gently and let sit for 5-10 minutes until the mixture becomes frothy.

2. Mix the dough: In a large mixing bowl, whisk together the flour, salt, and nutmeg (if using). In another bowl, combine the eggs, water, butter, and vanilla. Add the yeast mixture to the wet ingredients, then gradually stir in the dry ingredients until a dough forms. You may need to use your hands or a dough hook to knead it for a few minutes.

3. Knead the dough: Transfer the dough to a floured surface and knead for 5-7 minutes until smooth and elastic. Alternatively, use a stand mixer with a dough hook. If the dough is too sticky, add a bit more flour until it becomes smooth.

4. Let the dough rise: Shape the dough into a ball and place it in a lightly greased bowl. Cover with a damp cloth or plastic wrap and let it rise in a warm place for 1-1.5 hours, or until it has doubled in size.

5. Shape the doughnuts: Once the dough has risen, punch it down and roll it out on a floured surface to about 1/2 inch thick. Use a doughnut cutter or two round cutters (one large and one small) to cut out the doughnuts. Gather the scraps and re-roll as needed.

6. Heat the oil: In a deep pot or Dutch oven, heat about 3 inches of vegetable oil to 350°F (175°C). If you don’t have a thermometer, drop a small piece of dough into the oil – it should sizzle and rise to the top quickly.

7. Fry the doughnuts: Carefully place the doughnuts into the hot oil, cooking 2-3 at a time, for about 1-2 minutes per side until golden brown. Use tongs or a slotted spoon to flip the doughnuts and remove them from the oil. Let them drain on paper towels.

Make the Glaze:

8. Prepare the glaze: In a medium bowl, whisk together the powdered sugar, milk, vanilla extract, and a pinch of salt until smooth and glossy. If the glaze is too thick, add a bit more milk until it reaches your desired consistency.

9. Glaze the doughnuts: While the doughnuts are still warm, dip them into the glaze, covering the top. Let the excess glaze drip off before placing them on a cooling rack. Repeat with the remaining doughnuts.

Serve:

10. Serve immediately: The doughnuts are best served warm, right after glazing. Enjoy with a cup of coffee or hot chocolate!