Preheat oven to 425 degrees. Put oven rack in the middle.
Remove giblets from chicken.
Generously salt and pepper inside the cavity. Add the cut lemon and one whole sprig of rosemary inside.
Tie the legs together with kitchen twine.
Place chicken in a cast iron skillet or a roasting pan.
In a small bowl mix the soft butter, garlic, lemon zest and 1 tablespoon chopped fresh rosemary.
Gently pull the skin away from the chicken and massage the butter mixture all over the chicken under the skin, being careful not to break the skin.
Rub more butter on top of the chicken skin and sprinkle with more salt and pepper.
Cook chicken 45 - 55 minutes or until an instant thermometer inserted in the thickest part of the thigh registers 160 degrees.
If the chicken starts to get too brown on top, place a piece of foil over the top of the chicken and continue to cook.