In a small bowl, combine the yeast and warm milk. Stir gently and let stand for 5 minutes until the mixture becomes foamy, which means the yeast is active.

In a large mixing bowl, whisk together the flour, granulated sugar, salt, and nutmeg until evenly distributed. Create a well in the center of the dry ingredients.

Add the yeast mixture, eggs, melted butter, and warm water to the well. Stir with a wooden spoon until a shaggy dough forms, then knead in the bowl or on a floured surface for 5-6 minutes until smooth and elastic.

Place the dough in a lightly greased bowl, cover with a clean kitchen towel, and let rise in a warm spot for 45-60 minutes until doubled in size.

Turn the risen dough out onto a floured surface and roll to 1/2-inch thickness. Use a 3-inch round cutter for the donuts and a 1-inch cutter for the centers. Place cut donuts on a parchment-lined baking sheet, cover loosely, and rest for 10 minutes.

Heat 2-3 inches of vegetable oil in a heavy-bottomed pot or Dutch oven to 350-360°F. Use a thermometer to monitor the temperature throughout frying.

Gently slide 2-3 donuts into the hot oil without crowding the pot. Fry for 1-2 minutes per side until deep golden brown, flipping once when they naturally rise to the surface.

Remove donuts with a slotted spoon and drain on a wire rack set over paper towels. Let cool for 5 minutes before glazing.

Whisk together the powdered sugar, milk, and vanilla extract in a shallow bowl until completely smooth with no lumps. The glaze should coat a spoon but still drip slowly.

Dip the top of each slightly warm donut into the glaze, letting excess drip back into the bowl. Return to the wire rack and let the glaze set for 10 minutes before serving.