In a mixing bowl, combine the warm milk, active dry yeast, and 1 tablespoon of the granulated sugar. Stir gently and let it sit for about 5-10 minutes until it becomes frothy.
Add the melted butter and remaining tablespoon of sugar to the bowl. Mix well to combine all the wet ingredients.
In another bowl, whisk together 2 cups of bread flour and salt. Gradually add the flour mixture to the wet ingredients, stirring until a dough begins to form.
Transfer the dough onto a floured surface and knead for about 5-7 minutes until it becomes smooth and elastic. Shape it into a ball.
Place the dough in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1 hour or until it has doubled in size.
Once the dough has risen, punch it down to release the air. Roll it out on a floured surface to about ½-inch thick. Use a donut cutter to cut out the donuts and holes.
Cover the shaped donuts with the kitchen towel again and let them rise for another 30 minutes until they puff up.
In a deep fryer or heavy pot, heat vegetable oil to 375°F. Carefully drop the donuts into the hot oil, a few at a time. Fry for about 1-2 minutes on each side or until golden brown.
In a medium bowl, whisk together the powdered sugar, whole milk, and vanilla extract until smooth. Adjust consistency as needed.
While the donuts are still warm, dip the tops into the glaze, allowing the excess to drip off. Place the glazed donuts back on the cooling rack to set.
