In a large bowl, dissolve the yeast in warm water along with 1 teaspoon of the sugar. Let the mixture sit for 5–10 minutes until it becomes foamy.

Add the warmed milk, melted butter, egg, remaining sugar, and salt to the yeast mixture. Whisk to combine.

Gradually add in the flour, stirring until a soft dough begins to form. Turn the dough out onto a lightly floured surface and knead for about 5–7 minutes until smooth and elastic.

Place the kneaded dough in a lightly greased bowl. Cover with a clean kitchen towel or plastic wrap, and let it rise in a warm spot for about 1 hour or until doubled in size.

Gently punch down the dough to release air, then roll it out to about ½-inch thickness. Use a donut cutter (or a large round cutter and a small round cutter for the hole) to cut out the donuts. Place them on a parchment-lined baking sheet and let rise for another 30 minutes until slightly puffed.

Heat vegetable oil in a deep fryer or heavy pot to 350°F. Working in batches, carefully place the donuts in the hot oil, frying for 1–2 minutes per side until golden brown and cooked through. Remove with a slotted spoon and drain on a paper towel-lined plate.

While the donuts are cooling slightly, whisk together the powdered sugar, milk, and vanilla extract until smooth.

Dip each warm donut into the glaze, turning to coat fully. Place on a wire rack to set for a few minutes until the glaze firms up.