1.25 cups whole milk – divided; 1 cup warmed to 110°F (for dough)
0.33 cups granulated sugar
2.25 tsp active dry yeast
2 large egg
0.25 cups unsalted butter – melted and cooled
1.5 tsp vanilla extract
4 cups all-purpose flour
1.125 tsp fine salt
0.5 tsp ground nutmeg
2 quarts vegetable oil
2 cups powdered sugar
Instructions
In a large bowl, whisk whole milk (use 1 cup warm for the dough; reserve 0.25 cup for the glaze) with granulated sugar until dissolved, then sprinkle in active dry yeast; let stand until foamy, 5–10 minutes.
Whisk in eggs, unsalted butter, and vanilla extract (use 1 tsp now; reserve 0.5 tsp for the glaze). Stir in 3.5 cups of the measured all-purpose flour, fine salt (use 1 tsp now; reserve 0.125 tsp for the glaze), and ground nutmeg until a shaggy dough forms; knead until smooth and elastic, 6–8 minutes, adding the remaining measured flour only as needed to prevent sticking.
Form the dough into a ball, place in a clean bowl, cover, and let rise in a warm spot until doubled in size, 60–90 minutes.
Turn the dough onto a lightly dusted work surface and roll to about 0.5 inch thick. Cut rings with a 3-inch cutter and a 1-inch center cutter, re-rolling scraps once.
Arrange the cut dough on parchment-lined baking sheets, cover, and let rise until puffy and slightly springy to the touch, 30–45 minutes.
Heat vegetable oil in a large heavy pot to 350°F/175°C, at least 2 inches deep. Fry 2–3 doughnuts at a time until deep golden brown, 1–1.5 minutes per side, adjusting heat to maintain temperature.
Drain on a wire rack set over a sheet pan.
For the glaze, whisk powdered sugar with the reserved milk, the reserved vanilla, and the reserved salt until smooth and pourable; thin with a few drops of the reserved milk if needed.
While still slightly warm, dip each doughnut in the glaze, letting excess drip back, then return to the rack to set, 10–15 minutes. Serve the doughnuts fresh the day they are made.