Add quinoa, white beans, egg whites, minced garlic, chili powder, salt, and black pepper to a food processor. Process on high until thoroughly combined. It’s ready when it develops a thick consistency and nearly stops moving.
Remove from processor to a medium-sized bowl and combine with breadcrumbs and cheese. Mix thoroughly. Scoop large handfuls and form into balls. Flatten and make sure no cheese strands stick out.
Heat a frying pan on high heat until oil begins bubbling. Reduce heat to medium. Add the formed patties to the pan and cover. Let simmer for 2-3 minutes each side, or until browning. Add a slice of cheese to melt it, if desired. Remove from heat and add to sandwiches!