Make the dough base. In a large bowl, whisk the flour, sugar, yeast, and salt.
In a separate bowl, mix warm milk, eggs, and vanilla. Pour the wet ingredients into the dry and stir with a wooden spoon until shaggy.
Incorporate the butter. Add the softened butter and knead by hand on a lightly floured surface for 8–10 minutes, or use a stand mixer with a dough hook for 6–7 minutes on medium-low. The dough should be smooth, slightly tacky, and elastic.
If it’s very sticky, dust in 1–2 tablespoons of flour.
First rise. Place the dough in a lightly oiled bowl, cover, and let rise in a warm spot until doubled, about 60–90 minutes. A slow, steady rise builds better texture.
Shape the donuts. Turn the dough onto a floured surface and roll to about 1/2 inch (1.3 cm) thick. Use a 3-inch (7.5 cm) round cutter for the donuts and a 1-inch (2.5 cm) cutter for the holes.
Gather scraps, rest them 5 minutes, then re-roll to cut more. You’ll get about 10–12 donuts plus holes.
Second rise. Arrange cut donuts on parchment-lined trays, leaving space between them. Cover loosely and let rise until puffy, 30–45 minutes.
They should look airy and slightly jiggly if you nudge the tray.
Heat the oil. In a heavy pot or Dutch oven, pour oil to a depth of 2–3 inches. Heat to 350°F (175°C). Keep a thermometer clipped to the pot if possible.
Adjust the heat to hold a steady temperature.
Mix the coating. In a shallow bowl, combine the granulated sugar, cinnamon, and a pinch of salt. Stir well. Have this ready next to your frying station.
Fry the donuts. Carefully slide in 2–3 donuts at a time.
Fry for 1–1 1/2 minutes per side, until deep golden and crisp. Donut holes take about 60–90 seconds total. Transfer to a wire rack set over a sheet pan to drain for 20–30 seconds.
Coat while warm. While still warm but not dripping oil, toss each donut in the cinnamon sugar to coat all sides.
If the sugar starts clumping, refresh with a little extra sugar and a sprinkle more cinnamon.
Serve. Enjoy right away for the crispiest exterior and pillowy interior. These are best within a few hours, but you can refresh them (see Keeping It Fresh).