Warm the milk: Heat 3/4 cup of whole milk until it’s warm to the touch, about 105–110°F. If it feels hot, let it cool a minute.

Bloom the yeast: Stir 2 teaspoons active dry yeast and 1 tablespoon sugar into the warm milk. Let it sit 5–10 minutes until foamy. If it doesn’t foam, start over with fresh yeast.

Mix the wet ingredients: In a large bowl, whisk 1/4 cup melted unsalted butter (cooled), 1/4 cup sugar, 1 teaspoon vanilla, 1 large egg, and 1 egg yolk. Add the foamy yeast mixture and stir.

Build the dough: Add 2 3/4 to 3 cups all-purpose flour and 1/2 teaspoon salt to the bowl. Stir with a spoon or use a stand mixer with a dough hook. The dough should be soft and slightly tacky, not sticky.

Knead: Knead by hand on a lightly floured surface for 6–8 minutes (or 4–5 minutes in a mixer) until smooth and elastic. Add small sprinkles of flour only if it sticks.

First rise: Place dough in a greased bowl, cover, and let it rise in a warm spot until doubled, about 60–90 minutes.

Shape the doughnuts: Gently deflate the dough and roll it to about 1/2-inch thick. Cut out 3-inch rounds with a doughnut cutter or a round cutter and a small cap for the center hole. Re-roll scraps once if needed.

Second rise: Place cut doughnuts on parchment-lined trays. Cover lightly and let rise until puffy, 30–45 minutes. They should look airy and spring back slowly when touched.

Heat the oil: In a heavy pot, heat 2–3 inches of oil to 350°F. Keep a thermometer clipped on to maintain temperature.

Fry: Fry 2–3 doughnuts at a time, 45–60 seconds per side, until golden. Don’t overcrowd. Transfer to a wire rack set over a sheet pan. Fry holes too—they take about 30 seconds per side.

Make the coconut glaze: In a bowl, whisk 2 cups powdered sugar with 1/3 cup full-fat coconut milk until smooth and pourable. Add 1/4 teaspoon vanilla and a few drops of coconut extract if using. Adjust with more sugar to thicken or more coconut milk to thin.

Serve: Enjoy warm or at room temperature. The glaze firms up in about 20 minutes.