In the bowl of your stand mixer, place 2 1/2 cups flour, the xanthan gum, salt, cream of tartar, baking soda and sugar. Whisk to combine well. Add the yeast and optional nutmeg, and whisk again to combine well.

Fit the mixer with the paddle attachment.

Add the vinegar, egg, egg white, milk and melted butter, and mix to combine. Mix on low speed until the liquid is absorbed by the dry ingredients, then turn the mixer speed to high and mix for about 2 minutes or until very well-combined.

The dough will be wet but should scrape easily off the sides of the mixer with a spatula.

For best results, cover the inside of a lidded bucket or bowl with cooking oil spray, scrape the donut dough into the container, and cover the bucket or bowl.

Place in the refrigerator to rise and chill for at least 1 hour, or up to 2 days. You can work with the dough immediately, though, if you prefer. It will just be stickier and a bit harder to handle, and will have less flavor.

When you’re ready to work with the dough, turn it out onto a lightly floured surface. Line rimmed baking sheets with unbleached parchment paper and set them aside.

Sprinkle the dough lightly with some extra flour and turn over on itself a few times to create a smoother dough. Press the dough into a disk, sprinkle lightly with more flour, and roll it out about 1/2 inch thick.

Flour a doughnut cutter or biscuit or large round cookie cutter, and cut the dough into donut shapes. If you’re using a large round cutter, use a much smaller cutter to cut out donut holes from the rounds.

Place the shapes on the prepared baking sheets about 1 1/2-inches apart. Gather scraps and reroll, then cut more shapes.

Cover the baking sheets with oiled plastic wrap, and place in a warm, draft-free spot and allow to rise to about 150% of their original size.

Spray the bottom of the Air Fryer basket with non-aerosol cooking oil spray. Place as many donuts and holes as will fit comfortably in the basket of your Air Fryer in a single layer, without crowding.

Spray or brush the tops of the nuggets generously with cooking oil, and place in the Air Fryer. Set the machine to fry at 380°F for 12 minutes. Allow to cook for about 6 minutes.

Remove the basket carefully from the fryer and, using heat-safe tongs, flip each of the nuggets over. Spray or brush again generously with cooking oil, and return to the fryer.

Finish frying until lightly golden brown all over. Remove the donuts and place on a wire rack to cool. Repeat with the remaining donuts and holes.

In a large, heavy-bottom stock pot, heat at least 2 inches of oil to about 350°F.

Once the oil reaches temperature, fry a few old chunks of bread in the oil. They will blacken pretty quickly. Discard them.

Fry the doughnuts and holes in the hot oil in small batches, about 1 minute (or less) per side, until golden brown all over. Remove with a slotted spoon and drain on a wire rack lined with paper towels.

In a small-to-medium-sized bowl, place the confectioner’s sugar. Add the syrup or honey, and mix to combine into a thick paste.

Add water, a teaspoon at a time, and mix well until you have achieved a smooth and thickly pourable glaze.

Dip the top of each donut and donut hole in the glaze, allowing any excess to drip off, and then return to the wire rack to set, glazed side up.