Prepare the barley as the label directs. Let cool completely.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the onion, celery and a pinch of salt; cook, stirring occasionally, until golden, 12 to 14 minutes. Add the garlic, 3 tablespoons barbecue sauce and the carrot; cook, stirring, until the mixture dries out slightly, 1 to 2 minutes. Transfer to a food processor and let cool completely.
Add the barley, beans, breadcrumbs, walnuts, soy sauce, egg whites, parsley and 1/2 teaspoon salt to the food processor. Pulse until finely ground with some chunks. Form into six 4-inch-wide, 1/2-inch-thick patties and place on a baking sheet lined with parchment paper. Cover and refrigerate until firm, 1 to 4 hours.
Preheat the broiler. Heat the remaining 1 tablespoon olive oil in a large nonstick skillet over medium heat. Working in batches, cook the patties until golden brown, about 6 minutes per side. Meanwhile, place the buns, cut-side up, on a broiler pan and broil until toasted, 1 to 2 minutes. Serve the patties on the buns; top with the remaining 2 tablespoons barbecue sauce and onion rings.