In a large bowl, whisk together warm milk, yeast, and 1 tablespoon of the sugar. Let stand until foamy, about 5 minutes. Add remaining sugar, beaten egg, melted butter, flour, and salt. Stir with a wooden spoon until a shaggy dough forms, then turn out onto a lightly floured surface.

Knead the dough for 8-10 minutes until smooth, elastic, and slightly tacky but not sticky. It should pass the windowpane test - stretch a small piece thin enough to see light through without tearing. Add flour sparingly if needed, but a softer dough makes lighter donuts.

Place dough in a lightly oiled bowl, turn to coat, and cover with plastic wrap. Let rise in a warm spot until doubled in size, 60-75 minutes. The dough should hold a gentle indent when pressed.

Punch down dough and roll on a floured surface to about 1/2-inch thick. Cut with a 3-inch round cutter, then use a 1-inch cutter to remove centers. Gather scraps, re-roll once, and cut again. Place donuts and holes on floured baking sheets, cover loosely, and let rise 30-40 minutes until puffy and nearly doubled.

Heat oil to 350 degrees F in a heavy pot. Fry 2-3 donuts at a time, 60-90 seconds per side, until deep golden brown. Adjust heat to maintain temperature. Drain on a wire rack set over a baking sheet. Fry holes for 45-60 seconds total, rolling them occasionally.

Whisk confectioners' sugar, milk, and vanilla until smooth - the glaze should be thick but pourable, like honey. Dip warm donuts top-side down into glaze, let excess drip off, then return to rack glaze-side up. Let set 15 minutes before serving.