Activate the Yeast: In a large measuring jug, combine the milk and boiling water. Add a teaspoon of the sugar and the yeast. Stir gently, then leave in a warm place until the yeast activates and becomes foamy, about 5-10 minutes.
Prepare the Dough Base: In a large mixing bowl, combine the flour, remaining sugar, and salt. Cut in the butter using your fingers or a pastry blender until the mixture resembles coarse crumbs. This creates pockets of butter that will help develop a tender texture in your donuts.
Form the Dough: Add the lightly beaten egg and activated yeast mixture to the flour mixture. Mix thoroughly until a smooth dough forms, which typically takes about 5 minutes of continuous mixing. The dough should pull away from the sides of the bowl.
Knead and First Rise: Turn the dough out onto a lightly-floured counter and knead for 5-10 minutes until it feels springy and small bubbles form under the surface. Return the dough to the bowl, cover with a cloth or plastic wrap, and let rise in a warm place for about an hour until doubled in size.
Shape the Donuts: Once risen, place the dough on the counter and cut into 4 equal pieces. Working with one piece at a time, stretch it into a long rope about 1 to 1.5 inches wide. Cut the rope into 1-inch pieces, roll each piece into a ball with your hands, and place them on a baking tray or wire rack. Cover with a cloth and allow to rise while you heat the oil.
Fry the Donuts: Heat oil to 375°F (190°C). Carefully place the donut holes into the hot oil, frying only a few at a time to prevent overcrowding. Fry until golden brown on each side, about 2 minutes per side. Use a slotted spoon to turn them as needed.
Drain and Glaze: Remove the donuts from the oil and drain on paper towels or a wire rack. While the donuts are still warm, dip each one into the glaze, ensuring complete coverage. The warm donuts will help the glaze adhere properly and create that signature Krispy Kreme finish.
Prepare the Glaze: For the glaze, melt the butter and stir in the confectioners' sugar and vanilla. Add hot water, one tablespoon at a time, until the glaze reaches a smooth, pourable consistency. The glaze should be thin enough to coat the donuts but thick enough to set.