If using frozen turkey, thaw completely in the refrigerator (24 hours per 4-5 lbs). Remove giblets and neck from the cavity, then pat the turkey completely dry inside and out with paper towels. Let it sit at room temperature for 45-60 minutes before roasting.
Preheat oven to 325°F (165°C). Place the roasting rack in the pan and pour 2 cups of broth into the bottom of the pan to prevent the drippings from scorching.
In a small bowl, mix the softened butter with dried thyme, rosemary, sage, salt, and pepper until a uniform herb paste forms.
Gently loosen the skin from the breast meat and spread half the herb butter directly onto the meat underneath the skin. Rub the remaining butter all over the outside of the turkey, then drizzle and rub with olive oil and season with salt and pepper.
Loosely stuff the cavity with the quartered lemon, halved garlic head, and any remaining quartered onion pieces. Tuck remaining carrots, celery, and onion around the turkey in the pan.
Tie the legs together with kitchen twine and tuck the wingtips under the body to promote even cooking and a neater presentation.
Roast a whole 10-12 lb turkey for 2.5 to 3 hours (13-15 min per lb), or a 4-6 lb turkey breast for 1.5 to 2 hours (20 min per lb) at 325°F. Baste with pan juices every 45-60 minutes, and tent with foil if the skin browns too quickly.
Insert an instant-read thermometer into the thickest part of the thigh away from the bone — the turkey is done when it reads 165°F (74°C). Check the breast as well to confirm.
Transfer the turkey to a cutting board, tent loosely with foil, and rest for 20-30 minutes before carving. This allows the juices to redistribute for moist, flavorful meat.
Set the roasting pan over medium heat and whisk 3 tbsp flour into the drippings (with excess fat skimmed off) for 1-2 minutes. Slowly pour in 3 cups of broth while whisking constantly, then cook for 5-8 minutes until thickened. Season with salt and pepper.
Remove cavity aromatics and discard. Carve the turkey by removing the legs, separating the drumsticks from the thighs, and slicing the breast meat along the breastbone. Arrange on a platter with roasted vegetables and serve with gravy on the side.