In large mixing bowl, combine brown rice flour, potato starch, tapioca starch, dark brown sugar, baking powder, ground cinnamon, salt, xanthan gum, and ground nutmeg. Whisk to combine.

In small bowl, combine ground flax seeds and hot water. Stir. Allow to stand for 30 seconds. Add ground flax, apple juice, and oil to dry ingredients. Mix until a dough forms. Dough will be slightly sticky.

Generously white rice flour counter. Transfer dough to counter. Generously white rice flour top of dough. Pat dough into 1/2 inch thick rectangle.

Line baking sheet with paper towels and place granulated sugar in bowl. Set both aside.

Add enough oil to Dutch oven to measure depth of about 1 1/2 inches. Heat oil to 375°F.

Dip 2 1/2-inch cutter into white rice flour. Cut dough into rounds. Dip 3/4-inch cutter into white rice flour. Cut centers into doughnuts with small cutter.

Gently place doughnuts into hot oil. Fry until doughnuts rise to the top of the oil, about two minutes. Using the handles of two wooden spoons, carefully flip doughnuts. (If after two minutes doughnuts do not rise to top of oil, flip anyway.) Fry doughnuts an additional minute.

Using a spider (Chinese skimmer), remove doughnuts from hot oil. Drain for a few seconds on paper towels.

Transfer to granulated sugar. You want to coat the doughnuts in sugar while the doughnuts are still hot or the sugar won't stick. Repeat with remaining dough. Serve doughnuts immediately.

For doughnuts holes: Carefully place doughnut holes into hot oil. Fry for one minute. Turn, fry an additional minute.