Combine thyme, peppercorns, parsley, and bay leaf in a cheesecloth bag. Combine boiling water, chicken, herb bag, salt, and pepper in a large Dutch oven. Cover and let stand 10 minutes. Remove chicken and herb bag; set aside. Strain liquid; set aside.
Add 3 tablespoons water to Dutch oven; gradually add flour, stirring to blend thoroughly. Add chicken, herb bag, and reserved liquid. Cover and simmer for 30 minutes. Add mushrooms and onions to chicken mixture. Cover and simmer an additional 30 minutes.
Remove chicken to a serving platter; keep warm. Discard herb bag.
Combine yolks and lemon juice. Gradually stir a small amount of hot broth mixture into yolk mixture; add to remaining hot mixture, stirring constantly. Cook over low heat 2 minutes or until thickened. Pour sauce over chicken.