In a large bowl or plastic bag, combine brown rice flour, amaranth flour, potato starch, sugar, xanthan gum, yeast, and salt. Mix well and set aside.

Pour sourdough starter, water, and oil into the bread machine baking pan. Add eggs and egg we whites.

Select the Dough Cycle. As the bread machine is mixing, gradually add the dry ingredients, scraping bottom and sides of pan with a rubber spatula. Try to incorporate all the dry ingredients within 1 to 2 minutes. When mixing and kneading are complete, remove the kneading blade, leaving the bread pan in the bread machine. Quickly smooth the top of the loaf. Allow the cycle to finish. Turn off the bread machine.

Select the Bake Cycle. Set time to 60 minutes and temperature to 350°F. Allow the cycle to finish. Remove loaf from the pan immediately and let cool completely on a rack.