In a small bowl, dissolve the yeast in warm water. Let it sit for 5–10 minutes until foamy.

In a large bowl, combine the yeast mixture with warm milk, sugar, salt, butter, egg, and 2 cups of the flour. Mix until a dough forms.

Gradually add the remaining 1/2 cup of flour and knead until the dough is smooth and elastic (about 5–7 minutes by hand or 3–4 minutes with a mixer).

Place the dough in a greased bowl, cover, and let it rise in a warm spot for 1–1.5 hours, or until doubled in size.

Roll out the dough on a floured surface to about 1/2 inch thickness. Cut out doughnuts using a doughnut cutter or a round cutter with a smaller cutter for the center.

Place the cut doughnuts on a floured baking sheet, cover, and let rise again for 30–45 minutes.

Heat oil in a deep pan to 350°F (175°C). Carefully fry the doughnuts, a few at a time, for 1–2 minutes per side until golden brown.

Remove and drain on paper towels.

For the glaze, whisk together powdered sugar, milk, and vanilla extract in a bowl until smooth.

Dip each warm doughnut into the glaze and place on a rack to let the excess drip off and the glaze set.