Whisk mayonnaise, egg yolk, miso, chili powder, agave, paprika, and cayenne in a large bowl. Add cooled barley, mushrooms, beans, onion mixture, and cashew mixture. Mash with a potato masher until mixture is well blended and sticky, about 2 minutes. Transfer one-quarter of the mixture to food processor and pulse until very finely chopped. Return to bowl with remaining patty blend. Stir in cheddar and feta cheese; season with salt and pepper. Form mixture into 6 patties (lightly oiling your hands will prevent it from sticking).