In a small bowl, whisk together the warm milk, yeast, and 1 teaspoon sugar. Let the mixture stand until very bubbly, 5 to 10 minutes.

In the bowl of a stand mixer fitted with the dough hook, whisk together the melted butter, eggs, and remaining ¼ cup sugar until smooth. Stir in yeast mixture until combined. Add the flour and salt. Mix on low speed until a shaggy dough forms. Increase speed to medium-low, and knead until the dough is smooth and elastic, about 5 minutes. (The dough will be slightly sticky.)

Transfer the dough to a large, lightly oiled bowl. Cover and let rise in a warm, draft-free place until doubled in size, about 1 hour 15 minutes.

Line a large rimmed baking sheet with parchment paper and lightly dust with flour.

Punch down the risen dough. On a lightly floured surface, roll the dough into a ½-inch-thick rectangle. Using a donut cutter, cut out 12 donuts and 12 donut holes, re-rolling scraps as necessary. You can also use a 3-inch and 1-inch round cookie cutter to cut out the donuts. Place donuts and holes on the prepared baking sheet. Loosely cover and let rise in a warm, draft-free place, until doubled in size, about 15 to 20 minutes.

Preheat the air fryer to 350°F for 10 minutes. Spray the air fryer basket with cooking spray.

Working in batches, carefully place donuts in an even layer in the air fryer basket, about 1-inch apart. Lightly spray the tops with oil.

Cook for 6 minutes for donut holes and 8 minutes for big donuts or until golden brown. (I recommend frying the big donuts and donut holes separately since the cook times are slightly different.)

In a medium bowl, whisk together the confectioners’ sugar, milk, melted butter, and vanilla until smooth. Dip the warm donuts and donut holes into the glaze. Place on a wire rack and let the glaze set for about 5 minutes before serving. Store any leftovers in an airtight container for up to 3 days.