Make the burgers: Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until lightly browned and tender, 3 to 5 minutes. Remove to a plate. Add 1 more tablespoon oil to the skillet, then add the bell pepper. Cook, stirring, until lightly browned and tender, 2 to 3 minutes. Add the scallions and garlic and cook, stirring, until softened, about 1 minute. Add the beans, rice, 3/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until heated through, 3 to 5 minutes. Transfer to a large bowl and beat with a mixer on medium-high speed until the beans are mashed and the mixture is incorporated. Add the panko and beat to combine. Season with salt and pepper, if needed.
Form the mushroom-bean mixture into four 4- to 4 1/2-inch-wide patties (about 1/2 inch thick). Place on a plate lined with wax paper, cover and refrigerate until firm, at least 30 minutes.
Meanwhile, make the ranch sauce: Combine the mayonnaise, sour cream, buttermilk, vinegar and garlic powder in a small bowl. Stir until well combined. Stir in the dill and chives and season with salt and pepper, if needed. Refrigerate until ready to use.
Cook the burgers: Heat a large nonstick skillet over medium-high heat. Add the remaining 2 tablespoons olive oil. When hot, carefully add the patties and cook until browned, 4 to 5 minutes (reduce the heat slightly if the patties are browning too quickly). Carefully flip and cook until the other side is browned and the patties are hot in the center, about 3 more minutes, topping each patty with a slice of cheese during the last 2 minutes of cooking. Toast the buns, if desired. Serve the burgers on the buns with lettuce, tomato, cucumber and the ranch sauce.