Melt butter over medium heat. Add first 5 ingredients, sauté until slightly tender. 5 minutes. Add next 5 ingredients.
Meanwhile, in a large pot, combine broth, rice and salt. Bring to boil. Reduce heat and add celery mixture. Cover and simmer 30 minutes. Add more stock if too thick. Place egg yolks in small bowl, whisk in 1 cup hot soup, then whisk mixture back into soup. Add chicken and mushrooms. Discard bay leaves. Heat gently; do not boil.