To make the turkey: Position the rack in the lowest third of the oven and preheat to 400 degrees F.
Rinse the turkey and pat it dry with paper towels. Place the turkey on a rack set inside a large roasting pan. Place the orange and lemon wedges, onion, and 2 sprigs of each fresh herb in the main turkey cavity.
Tie the legs together to hold the shape of the turkey.
Stir 2 tablespoons of butter, the herbes de Provence, oil, and 1 1/2 teaspoons of each the salt and pepper in a small saucepan over medium heat just until the butter melts.
Rub the butter mixture all over the turkey and between the turkey breast meat and skin. Place the turkey neck and giblets in roasting pan. (Recipe can be prepared up to this point 1 day ahead. Cover and refrigerate. Let stand at room temperature 30 minutes before roasting.)
Cover the turkey breast with foil. Roast for 20 minutes.
Pour 3 cups of broth into the pan and stir to scrape up any brown bits on the bottom of the pan.
Add the remaining sprigs of fresh herbs to the pan.
Roast the turkey for 40 minutes. Reduce the oven temperature to 350 degrees F. Remove the foil from the turkey; pour 1 more cup of broth into the pan.
Continue roasting the turkey until a meat thermometer inserted into the thickest part of the thigh registers 165 degrees F to 175 degrees F or until the juices run clear when the thickest part of the thigh is pierced with a skewer, basting occasionally with pan juices, about 1 hour and 30 minutes longer. Transfer the turkey to a platter and tent with foil.
Let stand 30 minutes while preparing the gravy.
To make the gravy: Strain the turkey pan juices from the roasting pan through a sieve and into a 4-cup glass measuring cup; discard the solids.
Spoon off the fat from atop the pan juices. Add enough chicken broth, about 1 to 2 cups, to the pan juices to measure 4 cups total. Melt the remaining butter in a heavy large saucepan over medium-high heat.
Add the flour and whisk for 1 minute. Gradually whisk in the broth.
Simmer until the gravy thickens slightly, whisking often, about 10 minutes.