Combine chicken broth, water, onion, thyme sprigs, and celery stalk in a saucepan. Add chicken; bring to a simmer. Simmer gently for 15 minutes or until chicken is done. Remove chicken from pan; cool. Shred chicken. Strain cooking liquid through a sieve over a bowl, reserving 1 cup cooking liquid; reserve remaining cooking liquid for another use. Discard solids. Heat a skillet over medium-high heat. Add olive oil, shallots, and garlic; sauté 1 minute, stirring frequently. Add cremini mushrooms, kosher salt, and black pepper; sauté 3 minutes. Add wine; cook until liquid evaporates. Stir in flour; cook 1 minute. Stir in reserved cooking liquid; bring to a boil. Cook 1 minute. Remove from heat; stir in chicken and crème fraîche. Serve over toasted bread slices.