Yeast Activation: Add the yeast to a small bowl and pour in the warm water. Stir gently to mix, then allow it to rest until it becomes bubbly and frothy, typically taking around 5 minutes.
Mix Wet Ingredients: In the bowl of your stand mixer, combine the lukewarm milk, sugar, eggs, melted butter, and vanilla. Add the activated yeast mixture.
Add Dry Ingredients: Gradually add flour and salt, mixing on low speed until the dough starts to form.
Knead the Dough: Increase the speed to medium and knead until the dough is smooth and elastic, about 8 minutes.
First Rise: Transfer the dough into a bowl that's been lightly greased. Cover it with a clean cloth and set it in a warm spot to let it rise, allowing it to expand to twice its original size, which should take about an hour.
Shape Doughnuts: Roll out the dough to about ½-inch thickness. Cut out doughnuts using a doughnut cutter. Place cut doughnuts and holes on a floured surface, cover lightly, and let rise for another 30 minutes.
Frying: Heat oil in a deep-fryer or large pot to 375°F. Heat the oil to the appropriate temperature and fry the doughnuts in small batches. Each batch should cook until it's a rich golden brown, about one minute on each side. Transfer to a cooling rack.
Glazing: Mix powdered sugar, vanilla, and water to make the glaze. Dip each warm doughnut into the glaze, then return to the rack to set.