This recipe starts with the process used for Brown Chicken Stock, but wine is used in place of water. Its tannins and acids yield soft but profound flavors. Serve over mashed potatoes.
Preheat oven to 450°.
To prepare broth, arrange chicken drumsticks, carrots, and onion in a single layer in the bottom of 2 broiler pans or roasting pans coated with cooking spray. Bake at 450° for 30 minutes. Turn chicken over; rotate pans on oven racks. Bake 30 minutes or until browned.
Transfer chicken and vegetables to a 6-quart stockpot. Carefully discard drippings from broiler pans, leaving browned bits. Place 1 broiler pan on stovetop; add 2 cups wine to pan. Bring to a boil over medium-high heat. Reduce heat; simmer 2 minutes, scraping pan to loosen browned bits. Carefully pour contents of broiler pan into stockpot. Repeat procedure with remaining broiler pan and 2 cups wine.
Add remaining wine, thyme, parsley, and bay leaf to stockpot; bring to a simmer over medium-high heat. Reduce heat to low, and simmer gently 2 hours, skimming foam and fat from the surface occasionally. Strain broth through a fine sieve into a large bowl. Discard solids. Cover and chill broth for 8 hours or overnight. Skim solidified fat from surface of broth; discard fat.
To prepare chicken, heat a large saucepan over medium-high heat. Coat pan with cooking spray. Add chicken quarters to pan; cook 5 minutes on each side or until browned. Remove chicken from pan. Wipe drippings from pan with paper towels. Add chicken quarters, mushrooms, and broth to pan; bring to a boil. Cover, reduce heat, and simmer 10 minutes or until chicken is done. Remove chicken from pan; cover and keep warm.
Simmer broth mixture until liquid is reduced to about 1 1/2 cups (about 25 minutes). Combine flour and butter, stirring with a fork until a paste forms. Add flour mixture to pan, stirring with a whisk; simmer 4 minutes or until slightly thickened. Stir in salt and pepper. Spoon sauce over chicken; sprinkle with parsley.