Bring water and quinoa to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until quinoa is tender, 15 to 20 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Mash chickpeas roughly in a bowl. Add cooked quinoa, chicken, egg whites, beets, oat bran, onion, prunes, poblano pepper, salt, chile powder, cumin, and black pepper. Mix well. Form into 6 equal-size patties. Wrap in aluminum foil.
Bake in the preheated oven until no longer pink in the center and any juices run clear, about 20 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).