In a large bowl or bowl of a stand mixer (fitted with dough hook), combine the warm milk, 1 teaspoon sugar, and the yeast. Let sit for about 5-10 minutes, until foamy.
Meanwhile, melt the butter and lightly beat the egg.
Add remaining ¼ cup sugar, salt, and egg to the yeast mixture; stir or whisk to combine. Then mix in the melted butter and 2 cups of flour on low speed.
Scrape down the sides of bowl, then mix in remaining 1 cup flour. Dough should start to pull away from sides of bowl, but still be sticky. (Can add up to ¼ cup flour more, if needed.) Increase speed to medium-low and knead for 5 minutes. Dough will become more smooth and elastic, but still sticky.
Remove dough hook or transfer dough to a greased bowl, cover with plastic wrap and let rise until doubled. (With instant yeast, this will probably take about 30 minutes.)
Punch down dough and turn out on lightly floured surface. Gently roll out dough until a little less than ½-inch thick, between ¼-inch and ½-inch. Cut out as many donuts as you can with a round cutter, about 3 inches in diameter. Use a cutter about 1 inch in diameter to cut out the centers.
Transfer cut out dough to a lightly greased parchment paper or silicone mat. Cover loosely with greased plastic wrap. Let rise until doubled, about 20-30 minutes.
Preheat air fryer to 350˚F/177˚C. Lightly spray air fryer basket with oil spray. Carefully transfer donuts to basket in a single layer. Lightly spray donuts with oil spray and cook until lightly golden brown, about 4 minutes. Cooking time will vary depending on your air fryer and the thickness of your donuts, so keep an eye on it! Repeat with remaining donuts, then with the donut holes. Transfer donuts to a cooling rack.
Glaze: While donuts are in the air fryer, melt butter for the glaze in a medium bowl or saucepan. Stir or whisk in the powdered sugar and vanilla, until smooth. Stir or whisk in the hot water 1 tablespoon at a time until it reaches desired consistency, somewhat thin but not watery.
While donuts are still warm, but after cooling for a few minutes, submerge each into the glaze. For donut holes, glaze last and just dump them into the bowl and gently stir to coat.
Move cooling rack over the parchment paper used earlier or a piece of foil to make cleanup easier. Place the glazed donuts on the rack to allow excess to drip off and let sit until glaze hardens, about 10 minutes.