Combine porcini and 1 cup boiling water; cover and let stand 45 minutes.
Preheat oven to 425°.
Remove and discard giblets and neck from chicken. Trim excess fat. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Rub 1/4 teaspoon salt and 1/4 teaspoon pepper under loosened skin and over breast and drumsticks. Lift wing tips up and over back; tuck under chicken. Stuff thyme sprigs and onion quarters into body cavity. Tie ends of legs together with twine. Place chicken on a rack in a roasting pan. Bake at 425° for 1 hour or until a thermometer inserted into meaty part of thigh registers 165°. Remove chicken from pan; cover and keep warm.
Drain porcini in a colander over a bowl; reserve 2/3 cup liquid. Chop porcini.
Place a zip-top plastic bag inside a 2-cup glass measure. Pour chicken drippings into bag; let stand 10 minutes (fat will rise to the top). Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into a small saucepan, stopping before fat layer reaches opening; discard fat. Lightly spoon flour into a dry measuring cup; level with a knife. Add flour to drippings, stirring with a whisk until smooth; cook 1 minute over medium heat, stirring constantly. Add 1/3 cup mushroom soaking liquid and 1 1/2 cups broth, stirring with a whisk; cook 8 minutes or until thickened, stirring frequently with a whisk. Stir in 1/2 teaspoon salt; cover and keep warm.
Heat oil in a medium saucepan over medium-high heat. Add shallots to pan; sauté 3 minutes. Add porcini and button mushrooms; sauté 4 minutes or until tender. Stir half of mushroom mixture into gravy; cover and keep warm. Add remaining 1/2 cup broth, remaining 1/2 teaspoon salt, remaining 1/4 teaspoon pepper, remaining 1/3 cup mushroom soaking liquid, 1 cup water, and barley to remaining mushroom mixture in medium saucepan. Bring to a boil; reduce heat, and simmer 10 minutes or until barley is tender. Stir in 2 teaspoons chopped thyme and vinegar. Remove and discard skin from chicken. Serve chicken with stuffing and gravy.