Step 1: Make the Dough
Dissolve yeast in warm water; let sit 5–10 minutes until foamy.
In a large bowl, combine sugar, butter, warm milk, salt, and egg. Stir in yeast mixture.
Gradually add flour and mix to form a soft dough.
Turn dough onto a floured surface and knead for 5 minutes until smooth.
Place in a greased bowl, cover, and let rise in a warm spot for 1 to 1.5 hours, or until doubled.
Step 2: Shape the Donuts
Roll dough to ½-inch thickness on a floured surface.
Cut using a donut cutter or round cutters.
Place on a floured tray, cover, and let rise again for 30 minutes.
Step 3: Fry the Donuts
Heat 2–3 inches of oil to 350°F (175°C) in a deep pan.
Fry donuts in small batches for 1–2 minutes per side, until golden brown.
Drain on a paper towel-lined plate.
Step 4: Make the Glaze
In a bowl, whisk together powdered sugar, milk, and vanilla until smooth.
Step 5: Glaze the Donuts
While warm, dip each donut into the glaze, coating both sides.
Set on a wire rack and let the glaze set before serving.