Activate yeast mixture:

Combine yeast with a small amount of sugar in warm milk; let sit 5-10 minutes until frothy.

Make dough:

In a bowl, whisk flour and salt. Pour in yeast mixture, egg, remaining sugar, melted butter, and vanilla. Stir until a sticky dough forms. Turn onto a floured surface and knead 5-8 minutes until smooth and elastic.

First rise:

Set dough in a greased bowl, cover, and let rise in a warm spot until doubled, roughly 1 to 1 1/2 hours.

Cut doughnuts:

Press down dough gently and roll to 1/2 inch (1.5 cm) thickness. Stamp out rounds using a doughnut cutter.

Proof doughnuts:

Arrange doughnuts on parchment-lined sheets, loosely cover, and let proof 30-40 minutes until airy.

Bake doughnuts:

Heat oven to 350°F (175°C). Bake 10-12 minutes until the edges turn light gold.

Make glaze:

Move doughnuts to a cooling rack. While they cool, whisk confectioners' sugar with your chosen liquid to a smooth glaze.

Glaze doughnuts:

Submerge the top of each warm doughnut in the glaze. Allow to fully set before serving.