Place brown rice and vegetable broth into a small saucepan and place over high heat. Bring to a boil, lower heat, cover and allow to simmer until rice is fully cooked and water is absorbed, 40-45 minutes. Remove from heat and allow to sit, covered, 5 minutes more.

While rice cooks, coat the bottom of a large skillet with 2 tablespoons of olive oil and place over medium heat. Add onion and cook until softened, about 5 minutes. Push onions to the side of skillet and add mushrooms in an even layer. Allow to cook until tender and lightly browned, about 4 minutes, flip and cook until lightly browned on opposite sides, about 4 minutes more. Add garlic to skillet and flip everything a few times. Cook about 1 minute more and remove from heat.

Place cooked rice, mushroom mixture, beans, panko breadcrumbs, tahini, nutritional yeast, vinegar, flax seeds, thyme, salt and pepper into food processor bowl. Pulse everything a few times to mix. Do not overblend. Mixture should be moist, but firm enough to shape into patties. Add a bit more panko if mixture seems too moist.

Coat the bottom of a large skillet with a bit of olive oil and place over medium heat. Working in batches, shape mixture into patties and place in hot skillet, being careful not to overcrowd the skillet. Cook about 5 minutes on each side, until browned, lightly pressing down on the tops of patties, and then drawing the sides in with a spatula to help shape the patties during cooking.

Whisk all ingredients together in a small bowl, thinning with as much water as needed to achieve desired consistency.

Stuff patties into buns and top with toppings of choice.