Make turkey: Mix first 5 ingredients in small bowl. Pat turkey dry with paper towels and place on rack set in large roasting pan. If not stuffing turkey, place herb sprigs in main cavity. If stuffing turkey, spoon stuffing into main cavity. Tie legs together loosely to hold shape of turkey. Brush turkey with oil. Rub herb mix all over turkey. Place turkey neck and giblets in roasting pan. (Can be prepared 1 day ahead if turkey is not stuffed. Cover and refrigerate. Let stand at room temperature 1 hour before roasting.)
Position rack in lowest third of oven and preheat to 425°F. Drizzle melted butter all over turkey. Pour 2 cups broth into pan. Roast turkey 45 minutes. Remove turkey from oven and cover breast with foil. Reduce oven temperature to 350°F. Return turkey to oven; roast unstuffed turkey 1 hour (roast stuffed turkey 1 hour 30 minutes). Remove foil from turkey; pour remaining 2 cups broth into pan. Continue roasting turkey until meat thermometer inserted into thickest part of thigh registers 180°F. or until juices run clear when thickest part of thigh is pierced with skewer, basting occasionally with pan juices, about 1 hour 40 minutes longer. Transfer turkey to platter; tent with foil. Let stand 30 minutes. Reserve liquid in pan for gravy.
Meanwhile, prepare gravy: Mix flour and Sherry in small bowl until smooth paste forms. Melt butter in heavy large saucepan over medium-high heat. Add mushrooms and rosemary and sauté until mushrooms begin to soften, about 3 minutes. (Can be made 3 hours ahead. Cover flour paste tightly. Let paste and mushrooms stand at room temperature.)
Discard turkey neck and giblets from pan juices in roasting pan. Transfer pan juices to large glass measuring cup. Spoon off fat. Add enough chicken broth to measure 5 cups; add to saucepan with mushrooms. Add flour paste and whisk until smooth. Bring mixture to boil, stirring frequently. Boil until thickened to light gravy, about 10 minutes. Mix in cream, thyme and tarragon. Season with salt and pepper. Serve turkey with gravy.