Proof the yeast: In a large bowl, whisk together the warm milk, yeast, and 1 tablespoon of the sugar. Let stand for 5-10 minutes, or until the mixture is foamy.
Combine ingredients: Add the remaining sugar, salt, flour, melted butter, and egg to the yeast mixture. Stir until a dough forms, then turn it out onto a lightly floured surface and knead for 5-7 minutes, or until smooth and elastic.
First rise: Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for about 1 hour, or until doubled in size.
Shape the doughnuts: Punch down the dough and roll it out on a lightly floured surface to about 1/2-inch thickness. Use a doughnut cutter to cut out the doughnuts, then re-roll the scraps and cut out more doughnuts.
Second rise: Place the doughnuts on a lightly floured baking sheet and cover with a clean kitchen towel. Let rise for another 30 minutes, or until slightly puffy.
Fry the doughnuts: Heat about 2 inches of vegetable oil in a deep pot or Dutch oven to 350°F. Carefully drop a few doughnuts into the hot oil and fry for 1-2 minutes per side, or until golden brown.
Drain and glaze: Remove the doughnuts from the oil with a slotted spoon and transfer them to a paper towel-lined plate to drain. While still warm, dip the tops of the doughnuts in the glaze and let set.
