Preheat the oven to 475 ℉.
In a small bowl, mix butter, chopped rosemary, salt, pepper, and garlic together and set aside.
In the bottom of a Dutch oven, place sliced onions, potatoes, and carrots. Salt and pepper the vegetables well.
Clean the neck and giblets out of the chicken. Pat dry with paper towels. Use twine to tie the legs together.
Season the cavity generously with salt and pepper.
Use your hands to slather the butter mixture under the breast skin and all over the chicken.
Place in Dutch oven on top of sliced vegetables.
Roast in the oven uncovered for 15 minutes. Reduce the temperature to 350 ℉ and continue cooking uncovered. Approximately 20 minutes per pound or until the internal temperature reaches 165 ℉.
Remove from the oven and allow the chicken to rest for about 10 minutes before carving.
Carve the chicken and serve.