In a large mixing bowl or the bowl of a stand mixer, whisk together the warm milk, yeast, and granulated sugar. Let the mixture sit for about 5 minutes or until it becomes foamy. If the mixture doesn’t foam, start over with fresh yeast.

Add the flour, eggs, and salt to the mixture and mix with a dough hook on low speed for one minute until the ingredients are incorporated.

Scrape down the sides of the bowl and increase the speed to medium for 2 minutes until the dough starts pulling away from the sides.

Add the butter, one tablespoon at a time, and continue mixing until smooth and elastic. If the dough is still sticky, continue mixing until it feels soft and dry to the touch.

Lightly grease a large bowl, transfer the dough, and cover with greased plastic wrap. Let the dough rest in a warm place for 60 to 90 minutes or until it doubles in size.

Once the dough has risen, punch it down and transfer it to a floured surface. Roll the dough out to a thickness of about ⅜ inch.

Use a donut cutter (or a 3-inch round cutter and a ½ round cutter for the center hole) to cut out the donuts. Gather any scraps and re-roll them. You should have about 16 donuts and 16 donut holes.

Place the cut donuts on parchment-lined baking sheets, ensuring they have space to rise. Cover loosely with greased plastic wrap and let them rest for about 30 minutes.

Whisk together powdered sugar, milk, and vanilla extract in a medium bowl until smooth. Set aside for glazing.

Pour the vegetable oil into a large pot or Dutch oven and heat to 330°F to 350°F. Use a thermometer to ensure the oil is at the right temperature.

Carefully place the donuts in the hot oil, frying 3 to 4 at a time. Fry for 2 minutes per side until golden brown.

Use a spider strainer to lift the donuts out of the oil and place them on a metal cooling rack.

Once all donuts are fried, dunk them into the glaze on both sides and return them to the cooling rack. Let the glaze set for about 15 minutes before serving.