Dough Making: In your stand mixer bowl, combine flour, sugar, yeast, and salt. Add warm milk, beaten eggs, melted butter, and vanilla extract. Using the dough hook, mix at low speed until the dough comes together.
Kneading: Increase the speed to medium and knead until the dough is smooth and elastic, about 8 minutes.
First Rise: Place the dough in a greased bowl, cover with a damp cloth, and let it rise in a warm place until doubled in size, about 1 hour.
Shaping: On a floured surface, roll out the dough to about ½ inch thickness. Use a donut cutter to cut out donuts. Transfer to parchment-lined trays.
Second Rise: Cover the shaped donuts with a cloth and let them rise again until puffy, about 30 minutes.
Frying: Heat oil in a deep fryer or pot to 350°F (175°C). Fry the donuts in batches, cooking each side for about 2 minutes or until golden brown. Remove with a slotted spoon and drain on a wire rack.
Glazing: Mix powdered sugar, milk, and vanilla to make the glaze. Dip each warm donut into the glaze, then return to the wire rack to set.