Preheat oven to 425°. Mix mushrooms, potatoes, Brussels sprouts and garlic; drizzle with oil. Sprinkle with thyme, salt and pepper; toss to coat. Place in a shallow roasting pan. , Place chicken on a work surface, breast side down and tail end facing you. Using kitchen shears, cut along each side of backbone; discard backbone. Turn chicken over so breast side is up; flatten by pressing down firmly on breastbone until it cracks. Place chicken on a rack over vegetables. Twist and tuck wings under to secure in place. Combine butter and miso paste; spread over skin (mixture will be thick). , Roast until a thermometer inserted in thickest part of thigh reads 170°-175°, 1-1/2 to 1-3/4 hours, covering loosely with foil after 45 minutes of cooking. (Miso mixture on chicken will appear very dark while roasting.), Remove chicken from oven; tent with foil. Let stand 15 minutes before carving. If desired, skim fat and thicken pan drippings for gravy. Serve with chicken. If desired, top with additional fresh thyme.