Preheat oven to 350 degrees.
Remove neck and giblets from cavity of chicken and discard. Rinse chicken inside and out and pat dry with paper towels.
Place potatoes, carrots, and half of quartered and separated onion in the bottom of a 9x13 inch baking pan. Pour the chicken broth over the top of the vegetables.
Season the cavity of the chicken with salt and pepper. Stuff cavity with lemon, garlic, 2 sprigs of rosemary, and remaining half of the onion. Place the chicken over the top of vegetables in baking pan.
Loosen skin from the breast meat of the chicken by running a small spatula gently underneath the skin. Spread 2 tablespoons of softened butter underneath the skin on both breasts. Use your fingers to smush it around on top of the skin.
Season the outside of the chicken and vegetables with salt, pepper, and garlic powder. Melt the remaining tablespoon of butter and drizzle over the top of the chicken.
Make a little tent of aluminum foil and cover the breast meat of the chicken. (Leave the aluminum foil on for the first hour of roasting time then remove.) Roast your chicken for about 20 minutes per pound, or until a meat thermometer placed in the thickest part of the thigh reaches 165 degrees. Slice and serve with the roasted veggies, and the pan drippings ladled over the top.