In a large bowl, combine warm milk, yeast, and 1 tablespoon of sugar. Let it sit for 5-10 minutes until foamy.

Add the remaining sugar, eggs, melted butter, salt, and half of the flour to the yeast mixture. Mix well.

Gradually add the remaining flour, mixing until a soft dough forms. Knead on a floured surface for 8-10 minutes until smooth and elastic.

Place the dough in a greased bowl, cover, and let it rise in a warm place for 1-1.5 hours, or until doubled in size.

Punch down the dough and roll it out on a lightly floured surface to about 0.5 inch thickness. Cut out donut shapes using a donut cutter or two round cutters.

Place donuts on parchment paper, cover, and let them rise again for about 30 minutes.

Heat vegetable oil in a deep pot or Dutch oven to 350°F (175°C).

Carefully place donuts into the hot oil, a few at a time. Fry for about 1-2 minutes per side, until golden brown.

Remove donuts with a slotted spoon and place them on a wire rack lined with paper towels to drain.

While the donuts are still warm, whisk together powdered sugar, milk, and vanilla extract in a bowl until smooth. Adjust consistency with more milk or sugar as needed.

Dip each donut into the glaze, letting excess drip off, and place back on the wire rack to set.