In a small bowl, combine the yeast, warm water, and 1 tablespoon of sugar. Let it sit for 5–10 minutes until it gets frothy.
In a large mixing bowl, mix the yeast mixture, melted butter, warm milk, eggs, remaining sugar, and salt. Stir until combined.
Gradually add the flour, one cup at a time, mixing until you have a soft dough.
Turn the dough onto a floured surface and knead it for 5–7 minutes, or until it’s smooth and elastic.
Place the dough in a greased bowl, cover it with a damp towel, and let it rise in a warm place for about 1 to 1.5 hours until it doubles in size.
Punch down the dough and roll it out to about 1/2-inch thickness. Cut out doughnuts using a doughnut cutter or round cutters.
Place the cut doughnuts onto a parchment-lined baking sheet, cover lightly with a towel, and let them rise for another 30 minutes.
Heat about 2 inches of oil in a deep skillet or pot to 350°F (175°C).
Fry the doughnuts in batches for 1–2 minutes per side or until they are golden brown. Drain them on paper towels.
In a medium bowl, whisk the powdered sugar, milk, and vanilla until smooth.
Dip the warm doughnuts into the glaze, letting the excess drip off, and place them on a wire rack to set before serving.