In a food processor add 1 1/2 cups cooked quinoa and ingredients eggs through salt. Combine for 15 seconds or well incorporated.
In a medium mixing bowl, add the quinoa-bean mixture along with the remaining 1 cup of cooked quinoa and shredded cheese. Stir to combine and refrigerate for 20 minutes.
While quinoa mixture is chilling, prepare avocado sauce. In a blender or food processor, add all ingredients avocado through water, starting with 1/3 cup water and add 1 Tbsp at a time until desired consistency is reached. I added about 1/2 cup for a slightly thick sauce that is spreadable, add more or less to your preference. Set aside.
In a large skillet*, heat 1/2 tbsp oil or butter over medium heat (oil is divided since you will be cooking in 2 batches). Meanwhile, form 12 patties about a 1/2 in thick (I used a 1/4 measuring cup to divide mixture up evenly, then slightly flatten). Cook for about 6-9 minutes on each side or until browned and a crust is formed. Using a spatula, carefully flip over and cook an additional 6-9 minutes or until browned. Heat remaining 1/2 Tbsp oil or butter in pan and cook remaining patties for about 6-9 minutes on each side or until golden crust is formed.