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

Add the remaining sugar, beaten eggs, and melted butter to the yeast mixture. Stir well.

In a separate bowl, whisk together flour and salt. Gradually add the dry ingredients to the wet ingredients, mixing until a soft dough forms.

Turn the dough out onto a lightly floured surface and knead for 5-7 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 to about 0.5 inch thickness. Cut out donut shapes using a donut cutter or two different sized round cutters.

Place the cut donuts on parchment paper, cover loosely, and let them rise for another 20-30 minutes.

Heat vegetable oil in a large, heavy-bottomed pot to 350°F (175°C). Carefully place a few donuts into the hot oil, being careful not to overcrowd the pot.

Fry for 1-2 minutes per side, until golden brown. Remove with a slotted spoon and place on a wire rack to cool slightly.

While the donuts are still warm, whisk together powdered sugar, milk, and vanilla extract until smooth. If too thick, add a little more milk; if too thin, add more powdered sugar.

Dip each donut into the glaze, ensuring it's fully coated. Let the excess glaze drip off, and place back on the wire rack to set.

Serve and enjoy your homemade glazed donuts!