Set rack at lowest position in oven and preheat to 450° F. Rinse turkey inside and out; pat dry.
Starting at the neck end, slide a hand between the skin and the breast meat to loosen the skin. Loosen the skin on the thighs.
Mix the minced herbs into 8 tablespoons of softened butter. Rub 4 tablespoons of the herb butter under the skin of the turkey breast.
Gently press 3 cloves of garlic into each thigh, under the skin.
Sprinkle main cavity generously with kosher salt and pepper. Place 4 tablespoons of plain butter, all fresh herb sprigs, remaining garlic cloves, and lemon zest into main cavity.
Put 4 tablespoons of herb butter to the side and rub the rest over the outside of the turkey. Sprinkle outside generously with salt and pepper.
Place turkey on a rack, set in a large roasting pan. Pour one cup of broth into the bottom of the roasting pan. Tuck wing tips under the turkey.
Place turkey in oven, breast side up and roast for 20 minutes.
Reduce oven temperature to 350° F. Turn turkey breast side down and roast for 30 minutes.
Pour 3 cups of broth over the turkey and roast for another 30 minutes.
When finished, transfer turkey to platter and let rest for 30 minutes (internal temp should rise 5-10° F).
Strain pan juices into a bowl.
Melt reserved herb butter in a large, heavy saucepan over medium heat.
Add flour and stir until roux is golden brown, about 6 minutes.
Gradually add pan juices; increase heat and whisk constantly until gravy thickens, boils, and is smooth.