Activate the Yeast: In a small bowl, dissolve the yeast in warm water with a teaspoon of sugar. Let it sit for 5-10 minutes until frothy.
Mix Wet Ingredients: In a stand mixer, combine the activated yeast, lukewarm milk, sugar, melted butter, and eggs. Mix until well incorporated.
Add Dry Ingredients: Gradually add the flour and salt to the wet mixture. Using the dough hook, knead on medium speed until the dough is smooth and pulls away from the sides of the bowl, about 10 minutes.
First Rise: Place the dough in a greased bowl, cover with a damp cloth, and let it rise in a warm area until doubled in size, about 1 hour.
Shape Donuts: On a floured surface, roll out the dough to about ½ inch thickness. Cut out donuts using a donut cutter. Place cut donuts and holes onto parchment paper.
Second Rise: Cover the shaped dough with a light towel and let rise for another 30 minutes.
Fry the Donuts: Heat oil in a deep fryer or large pot to 375°F. Fry the donuts in batches, until golden brown, about 1 minute per side. Use a slotted spoon to transfer to a wire rack.
Glaze: Mix powdered sugar, milk, and vanilla in a bowl until smooth. Dip each warm donut into the glaze, then return to the wire rack to set.