In your standing mixer with the dough hook attached, combine 2 ½ cups of the flour and yeast and whisk together. (You can use a mixing bowl and wooden spoon, if you don’t have a standing mixer.)
Heat milk, butter, sugar, and salt in a small saucepan stirring all the while until butter almost melts and mixture reaches 110-115 ⁰ F. (Do NOT boil.) Add milk mixture to dries and stir to incorporate with a wooden spoon.
Add eggs and beat by hand or with electric mixer on low speed for about 30 seconds. Beat for another three minutes on high speed, scraping down the sides of the bowl with a rubber spatula.
Add enough of the remaining flour, (half cup at a time), and mix by hand to form a soft dough.Place the dough on a lightly floured surface and knead for about 5-8 minutes. You may need to add a little bit of flour as you knead the dough, and you’ll need to sprinkle the work surface with extra flour, too, as you go.When it’s ready, the dough should be slightly elastic, and bounce back when you pull at it. Shape the dough into a smooth ball and place it in a large greased bowl, turning once to lightly grease the top of the dough. Cover top of the bowl with a slightly damp clean kitchen towel, and place in a warm, or sunny spot (NOT in a heated oven), for about 1¼ hours, or until the dough has doubled in size. When the dough has risen, punch a little hole in it to let it deflate.
On a floured work surface, roll dough out to a thickness of 1/2-inch. Cut out circles using a 3 or 4-inch biscuit cutter or a glass with a similar diameter. (They also make donut cutters, which would be nice, but we don’t have one!) Place the dough rounds on a board lined with parchment paper. Gather any leftover dough and roll it out again, using every bit of dough for your donuts.
Using a small one-inch (or so) biscuit cutter or a narrow shot glass, cut out the center of each round to create the classic donut shape and the donut holes.
Allow the donuts and donut holes to rise again for about 30 minutes.
Preheat air fryer to 345° F. Place a wire cooling rack on top of a baking sheet. When the air fryer is ready, lay 4 donuts at a time in the basket of the air fryer and bake for 5 minutes. They should be golden-brown and tender to the touch, but no longer doughy. Remove from fryer and place on the wire rack.
As soon as you can handle the donuts, dip them in the glaze and put them back on the wire rack. We like to dip the donuts twice for extra good coverage. Sprinkle immediately with whatever fun sprinkles you like.
Repeat this process until all donuts and donut holes are baked and glazed. For glazing the donut holes, just pop them into the bowl of glaze and gently toss them to cover each.