Activate the Yeast: In a small bowl, dissolve the yeast in warm water with a pinch of the sugar. Let it sit for about 5-10 minutes, until the mixture becomes frothy.
Make the Dough: In a large mixing bowl, combine the remaining sugar, melted butter, warm milk, egg, and salt. Add the yeast mixture and stir until well combined. Gradually add the flour, one cup at a time, mixing until a soft dough forms.
Knead the Dough: Turn the dough out onto a lightly floured surface and knead for about 5-7 minutes, or until the dough is smooth and elastic. Place the 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 it has doubled in size.
Shape the Doughnuts: Once the dough has risen, punch it down to release the air. Roll the dough out on a floured surface to about 1/2 inch thickness. Use a doughnut cutter or two round cutters (one large and one small) to cut out doughnut shapes. Re-roll the scraps to cut out more doughnuts.
Second Rise: Place the cut doughnuts on a lightly floured baking sheet, cover them with a kitchen towel, and let them rise for another 30-45 minutes, or until they are puffy.
Heat the Oil: In a large, deep pot, heat vegetable oil to 350°F (175°C). The oil should be deep enough to submerge the doughnuts.
Fry the Doughnuts: Carefully place a few doughnuts at a time into the hot oil. Fry for about 1-2 minutes per side, or until they are golden brown. Use a slotted spoon to remove them from the oil and place them on a paper towel-lined tray to drain any excess oil.
Make the Glaze: While the doughnuts are cooling slightly, prepare the glaze by whisking together the powdered sugar, milk, and vanilla extract in a bowl until smooth.
Glaze the Doughnuts: Dip each warm doughnut into the glaze, coating it completely. Place the glazed doughnuts on a wire rack set over a baking sheet to allow the excess glaze to drip off.
Serve: Allow the glaze to set for a few minutes before serving. Enjoy your homemade Krispy Kremes warm for the best taste!