Prepare the dough: In a stand mixer, blend flour, xanthan gum, tapioca starch, salt, cream of tartar, baking soda, and sugar until well-combined. Add yeast and nutmeg, mix again. Add vinegar, egg, egg white, milk, and butter. Mix on low until combined, then on high for 2 minutes.
Chill and rise: Transfer dough into an oiled bowl, cover, and refrigerate for at least 1 hour to rest. It can stay up to 4 days in the fridge!
Shape the donuts: Roll the dough onto a floured surface to ½-inch thickness. Use a donut cutter to create shapes, then lay them on parchment-lined trays.
Final rise: Cover the trays and allow donuts to rise for 45-60 minutes until 1 ½ times their original size.
Air fry or deep fry: For air frying, cook at 380°F for 12 minutes, flipping once. Spray generously with oil. For deep frying, cook in 2 inches of oil heated to 350°F for 2-3 minutes per side.
Glaze: Mix confectioner’s sugar, golden syrup, and water into a thick glaze. Dip warm donuts and set on cooling racks.