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Patties: Preheat oven to 350°. Cook barley in a small pot of boiling salted water until very tender, 40–50 minutes. Drain; let cool.
Meanwhile, chop mushrooms into ¼"–½" pieces. Arrange in an even layer on a large rimmed baking sheet. Drizzle with 1 Tbsp. oil, season with salt and pepper, and toss to coat. Roast until mushrooms have lost half of their volume and are tender, 12–15 minutes. This step is essential—it removes much of the excess liquid from the mushrooms, which would lead to a mushy patty. Let cool.
Spread beans on another large rimmed baking sheet and roast, tossing once, until most of them have split open and look dry around the edges, 8–10 minutes. The beans are an essential addition for binding the patty, and add protein. Let cool.
Meanwhile, heat remaining 1 Tbsp. oil in a medium skillet over medium. Cook onion and garlic, stirring frequently, until soft, 8–10 minutes. Let cool. Pulse cashews and oats in a food processor until finely chopped.
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 Cotija cheese; season with salt and pepper. Form mixture into 6 patties (lightly oiling your hands will prevent it from sticking).
Do Ahead: Patties can be made 2 days ahead; wrap tightly and chill. Or, freeze in a single layer, then wrap tightly up to 3 months. Thaw before cooking.
Assembly: Heat oil in a large cast-iron skillet over medium. Working in batches, cook patties until dark brown on both sides, about 10 minutes total. Place cheese slices over burgers, if using; cover and cook just long enough for cheese to melt, about 1 minute. Serve on buns with your choice of toppings.
3 hours, 30 minutes