Combine mustard, 3 teaspoons thyme, pepper, and garlic in a small bowl.
Rinse chicken with cold water, pat dry. Trim excess fat. Starting at neck cavity, loosen skin from breast by inserting fingers between skin and meat. Rub mustard mixture under skin. Lift wing tips out and tuck under chicken. Brush oil over chicken skin.
Place chicken in a large ovenproof skillet. Pour 1 1/2 cups broth around chicken; bake for 1 hour or until juices run clear.
Remove chicken from skillet. Remove and reserve broth mixture. Skim fat from surface of broth; discard. Combine reserved broth mixture, 1 cup broth, and flour, stirring with a whisk until blended. Heat skillet over medium-high heat; add vermouth and cook 2 minutes. Add broth mixture, bring to a boil; cook 1 minute or until slightly thick, stirring constantly. Stir in 1/2 teaspoon thyme and salt.
Note: Vermouth is a fortified white wine flavored with a variety of herbs and spices. You may, however, use fat-free, less-sodium chicken broth in its place.
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