Activate the Yeast: Combine warm milk and sugar in a small bowl. Sprinkle the yeast over the top and let it rest for 5–10 minutes until it becomes frothy.
Make the Dough: In a large bowl, combine flour and salt. Add the activated yeast mixture, melted butter, egg, and vanilla extract. Mix until a soft dough forms.
Knead the Dough: Transfer the dough to a floured surface. Work the dough for 8–10 minutes until it is smooth and stretchy.
Let the Dough Rise: Place the dough in a lightly greased bowl, cover with a damp towel, and let it rise in a warm spot for 1–2 hours or until it doubles in size.
Shape the Doughnuts: Flatten the dough on a floured surface to approximately ½-inch thick. Use a doughnut cutter or two round cutters (one for the outer circle and one for the hole) to shape the dough.
Second Rise: Place the shaped doughnuts on parchment paper. Cover them with a towel and let them rise for 30–45 minutes until puffed.
Fry the Doughnuts: Warm vegetable oil in a deep pot until it reaches 350°F. Fry 2–3 doughnuts at a time for about 1–2 minutes per side, until golden brown. Remove and drain on a wire rack.