Add quinoa and stock to a medium sauce pan and bring to a boil. Reduce to a simmer and cover, cooking for 15 minutes or until can be fluffed with a fork. You can do this ahead of time or even the night before (and store quinoa in the fridge) so the quinoa can cool.
While quinoa is cooking, slice the green onions, chop the carrots, mince the garlic and grate the cheese. Place it all in a bowl, then add panko and mix. Coarsely chop the chickpeas and add to the bowl. One quinoa is finished cooking, add it to the bowl and mix well until incorporated. If needed, let cool or place in the fridge for a few minutes to cool. Add in beaten egg + egg white, salt, pepper and mix well. Form into patties (I really squeezed a handful of quinoa together and smoothed it into a round, it may take some squeezing and form to get the patties to stick) that fit the size of the buns you have. <- this will differ!
Heat a large skillet over medium-high heat and add 1 1/2 tablespoons olive oil. Once hot, add 5-6 quinoa patties and cook until deeply golden brown and crispy, about 5 minutes. Gently flip and cook 5 minutes more. Remove and continue with remaining olive oil and patties. Serve with sliced avocado chipotle mayo!