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.

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.

Transfer the dough to a lightly greased bowl, cover with greased plastic wrap, and let it 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 to cut out the donuts and gather any scraps to re-roll.

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

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

Heat vegetable oil in a large pot or Dutch oven to 330°F to 350°F.

Carefully fry the donuts in batches of 3 to 4 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 cooling rack.

Dunk the fried donuts into the glaze on both sides and return them to the cooling rack. Let the glaze set for about 15 minutes before serving.