• Cinnamon sugar, for sprinkling, optional
Heat the oven to 325 degrees, with a rack in the middle. Brush two six-mold doughnut trays with coconut oil or butter.
Whisk together the sugar, rice flour, bean flour, potato starch, arrowroot, baking powder, xanthan gum, salt and baking soda. Add 1/3 cup of oil or butter, applesauce, vanilla and 1/2 cup hot water and mix with a rubber spatula until the ingredients are just combined. Be careful not to overmix. Drop about 21/2 tablespoons batter into each doughnut mold and spread evenly around with a toothpick.
Bake for 8 minutes, turn the pans around and bake until the doughnuts are golden brown -- about 7 more minutes. Cool in the molds for 5 minutes. Run a knife around the doughnuts to lift them out of the molds, and place on a baking sheet. Sprinkle with cinnamon sugar and eat warm or at room temperature.
Nutrition information per doughnut: