Add the remaining beans, shallot, garlic, spices and hot sauce into the bowl of a food process fit with a steel blade. Process until the mixture is the texture of a very chunky puree. Transfer this mixture to a mixing bowl.
Add the remaining beans, cooked quinoa and flax egg and stir together to form your ‘dough’. If the dough feels too sticky and isn’t holding together as you shape the patties, add some gluten-free flour 1 tablespoon at a time, until they start to are easier to shape (but will still be sticky!).
Shape the mixture into 8 patties and chill in the fridge for at least 1 hour, more if you have time.
Preheat oven to 375ºF. Line a baking sheet with parchment paper. Transfer burgers to parchment and bake for 30 - 40 minutes, flipping halfway through until both sides are browned.