Preheat oven to 400°F. Bring 4 cups water and 1 tablespoon salt to a boil in a medium saucepan over medium-high. Add barley and bay leaf; return to a boil. Reduce heat to medium-low, and simmer, undisturbed, until barley is al dente, 25 to 30 minutes. Drain well, and transfer to a large bowl; remove and discard bay leaf. Let cool slightly, about 15 minutes.

While barley cooks, toss together mushrooms, tamari, 2 tablespoons oil, pepper, and remaining 1/4 teaspoon salt on a large rimmed baking sheet; spread in an even layer. Place garlic head halves on a sheet of aluminum foil, and drizzle with 1 teaspoon oil. Tightly wrap foil around garlic, and place on baking sheet next to mushrooms. Roast in preheated oven until mushrooms are browned and beginning to dry out, about 40 minutes, stirring twice during cook time. Remove from oven; let cool on baking sheet 10 minutes. (Do not turn oven off.) Remove and discard foil from garlic; squeeze garlic cloves from skins, and set aside. Discard garlic skins.

Transfer cooled barley to a food processor; pulse until roughly chopped, about 15 pulses. Transfer to a large bowl; add carrots, beets, cumin, and paprika. Transfer roasted mushrooms to food processor; pulse until finely chopped, about 10 pulses. Transfer mushrooms to barley mixture. Add chickpeas to food processor; pulse until roughly chopped, about 10 pulses. Transfer 1 cup chopped chickpeas to barley mixture. Add roasted garlic cloves to remaining chickpeas in food processor; process until smooth, about 1 minute, stopping to scrape down sides of bowl as needed. Transfer chickpea-garlic mixture to barley mixture.

Add panko, walnuts, and mustard to barley mixture, and stir until well combined. Shape mixture into 8 (1/2-inch-thick) patties (about 1/2 cup each).

Heat 2 tablespoons oil in a large skillet over medium. Add 4 patties; cook until golden brown and crispy on both sides, about 2 minutes per side. Transfer seared patties to a large baking sheet. Repeat process with remaining 2 tablespoons oil and remaining 4 patties. Bake patties at 400°F until tops and sides are crispy, about 10 minutes. Serve on buns with mayonnaise, lettuce, pickles, red onion, tomato, and vegan cheese, if desired.