In a large mixing bowl, combine the warm milk (107-110°F), active dry yeast, and 1 tablespoon of the granulated sugar. Stir gently and let it sit for about 5-10 minutes until foamy.
Once the yeast is frothy, add the melted butter, remaining tablespoon of sugar, and salt to the mixture. Gradually add 2 cups of bread flour, stirring until a dough begins to form. Add more flour if dough is too sticky.
Turn the dough out onto a floured surface and knead for about 5-7 minutes, or until smooth and elastic. The dough should be slightly tacky but not overly sticky.
Place the kneaded dough in a lightly greased bowl, cover with a clean kitchen towel, and let it rise in a warm place for about 1-2 hours, or until doubled in size.
Punch down the risen dough. Roll it out on a floured surface to about 1/2 inch thick. Cut out donuts using a donut cutter or two round cookie cutters. Place on a floured baking sheet and let rise again for 30-45 minutes.
Heat vegetable oil in a deep frying pot or large saucepan to 350°F. Use a thermometer to check the temperature.
Carefully fry the donuts in batches for about 1-2 minutes per side, or until golden brown. Remove with a slotted spoon and place on a cooling rack to drain excess oil.
In a medium mixing bowl, whisk together powdered sugar, whole milk, and vanilla extract until smooth. Add more milk if glaze is too thick.
Dip the slightly cooled donuts into the glaze, allowing excess to drip off. Place back on the cooling rack to let the glaze set.