Remove the giblets and neck from chicken, discarding the liver. Cut off the wing tips, and combine with giblets, neck, 3 cups broth, carrot, celery, onion, garlic, and bay leaves in a saucepan. Bring to a boil; partially cover, reduce heat, and simmer for 45 minutes. Strain broth mixture through a sieve into a bowl, reserving broth and giblets. Mince giblets; set aside. Discard the remaining solids.
Preheat oven to 375°.
Rinse chicken with cold water; pat dry. Trim excess fat. Starting at the neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Combine salt, thyme, and sage in a small bowl. Rub thyme mixture under loosened skin and over breast and drumsticks. Gently press skin to secure. Tie legs with string.
Place chicken, breast side up, on the rack of a broiler pan or roasting pan. Insert a meat thermometer into meaty part of thigh, making sure not to touch bone. Bake at 375° for 1 hour and 30 minutes or until the thermometer registers 180° and juices run clear. Cover chicken loosely with foil, and let stand 10 minutes for chicken to reabsorb juices. Discard skin. Remove rack from pan. Add 1/2 cup broth to drippings in pan (you'll have about 3/4 cup drippings), scraping pan to loosen browned bits.
Pour drippings into a zip-top plastic bag. Snip off a corner of bag; drain liquid into saucepan, stopping before fat layer reaches opening. Discard fat.
Add the reserved giblet mixture to the broth mixture in saucepan.
Combine remaining 1/2 cup broth and flour in a bowl. Add flour mixture to saucepan; bring to a boil. Reduce heat, and cook 10 minutes, stirring constantly. Stir in reserved giblets and pepper. Serve with chicken.
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