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.