In a small bowl, combine the ground flaxseed and water, stirring until well mixed. Set it aside for a few minutes to thicken; this will serve as your flax egg substitute.
Place the drained black beans into a large mixing bowl. Using a potato masher or fork, mash the beans until most are broken down into a paste, leaving around 25% of them partially intact for texture.
In a food processor, pulse the cashews until they resemble coarse breadcrumbs. Be careful not to grind them into a fine powder—larger pieces are perfectly fine. Alternatively, chop them with a knife. Transfer the chopped nuts to the bowl with the beans.
Add the flaxseed mixture to the bean and nut mixture, along with the cooked rice or quinoa, shredded carrots, chopped parsley, green onions, breadcrumbs, smoked paprika, chili powder, and salt. Mix everything thoroughly using a large wooden spoon until well combined.
Scoop about 1/2 cup of the mixture for each burger patty and shape into patties that are approximately 3/4 inch thick.