Prepare Yeast: Warm ½ cup of milk to about 110°F, sprinkle 2 teaspoons of yeast over it, and let sit for 5 minutes until foamy.

Mix Dough: In a bowl, combine ¼ cup sugar, 2 tablespoons melted butter, 1 egg, 3 ½ cups flour, and 1 teaspoon salt. Mix in yeast mixture until shaggy dough forms.

Knead Dough: Knead dough on a floured surface for 5-7 minutes until smooth and elastic.

First Rise: Shape into a ball, place in a greased bowl, cover, and let rise in a warm spot for about 1 hour until doubled.

Shape Doughnuts: Punch down the dough, roll to ½ inch thick, cut out doughnuts and holes, and place on parchment-lined tray.

Second Rise: Cover shaped doughnuts and let rise for 30 minutes until puffed.

Fry Doughnuts: Heat oil to 350°F and fry doughnuts for 1-2 minutes on each side until golden brown.

Prepare Glaze: Whisk together 2 cups powdered sugar, ¼ cup milk, and 1 teaspoon vanilla until smooth.

Glaze Doughnuts: Dip warm doughnuts into glaze, allow excess to drip off, and place on wire rack to set.