To a small mixing bowl add the melted butter, garlic, lemon zest, 1/2 teaspoon fresh thyme, and a generous pinch of salt and freshly cracked pepper.
Place the celery and carrots in a cast iron and skillet. Use a paper towel to pat the chicken dry and then place the chicken on top of the veggies, breast side up.
Rub the melted butter mixture under the skin of the chicken as well as inside the cavity of the chicken, being sure to use all of the butter.
Stuff the chicken cavity with the head of garlic, pieces of yellow onion, lemon, and fresh thyme. If you like, use a bit of kitchen twine to tie the drumsticks.
Drizzle the olive oil over the skin of the chicken and then use a basting brush to ensure that the skin is well coated with oil. Liberally season the chicken with salt and fresh cracked black pepper.
Place the chicken in the oven, on the middle rack, and roast for 1 hour and 15-30 minutes, basting every 30 minutes, or until the internal temperature, at the thickest part, reaches 165 degrees.
Remove the chicken from the oven and let rest for 10-15 minutes in the skillet.
Carve the chicken and serve with the pan drippings or reserve the pan drippings to make homemade chicken gravy.