Prepare the dough:

• In a large bowl, dissolve the yeast in the warm milk and let it sit for 5-10 minutes until foamy.

• Add sugar, salt, vanilla extract, ground nutmeg, melted butter, and eggs to the yeast mixture. Stir to combine.

• Gradually add the flour, about 1/2 cup at a time, until the dough begins to come together. Once the dough is formed, add the warm water and mix until the dough is soft and slightly sticky.

• Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until it’s smooth and elastic.

• Place the dough in a lightly oiled bowl, cover with a clean cloth, and let it rise in a warm place for 1-1.5 hours or until it has doubled in size.

Shape the doughnuts:

• Once the dough has risen, punch it down and turn it out onto a floured surface.

• Roll the dough to about 1/2 inch thickness using a rolling pin.

• Using a round cookie cutter or drinking glass, cut out doughnut shapes. Use a smaller cutter or bottle cap to cut out the center of each doughnut to create the hole.

• Place the cut doughnuts and doughnut holes on a parchment-lined baking sheet, cover with a cloth, and let them rise for another 30 minutes.

Fry the doughnuts:

• Heat the vegetable oil in a deep fryer or large pot to 350°F (175°C). Test the oil by dropping a small piece of dough in; if it bubbles and floats to the surface, it’s ready.

• Carefully place the doughnuts into the hot oil, a few at a time, without overcrowding. Fry for 1-2 minutes per side, or until golden brown.

• Use a slotted spoon to remove the doughnuts from the oil and place them on a paper towel-lined plate to drain excess oil.

Make the glaze:

• In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth and free of lumps. Adjust the consistency by adding more milk if needed.

Glaze the doughnuts:

• While the doughnuts are still warm, dip them into the glaze, covering the top completely. Let the excess glaze drip off and allow the glaze to set for a few minutes.

Serve and enjoy:

• Serve the doughnuts immediately, or let them cool slightly. They’re best enjoyed fresh and warm but can be stored in an airtight container for up to 2 days.