Bring a large pot of water to a boil over high heat. Add crayfish, cook for 5 minutes, then drain and run under cold water. Remove meat from tails, reserving shells. Devein crayfish as you would shrimp: Run a paring knife along back of tail and lift out and discard intestine. (This is easiest under running water.) Set meat aside.
Heat oil in a large pot over medium-high heat. Add shells and cook, stirring, for 3 minutes. Add cognac, then carefully ignite cognac with a long-handled match. (Keep lid handy so flame can be extinguished if necessary.) Allow alcohol to burn off, about 1 minute. When flame has died down, reduce heat to medium and add celery, carrots, and garlic. Cook, stirring, until vegetables are soft, 10-15 minutes.
Reduce heat to medium-low, stir in tomato paste, and cook, stirring, for 2 minutes. Add stock, thyme, bay leaves, and 1 cup water. Simmer for 30 minutes.
Strain soup through a fine sieve, discarding shells, herbs, and vegetables. Return soup to pot and simmer over medium heat for 15-20 minutes. Ladle 1⁄2 cup of soup into a small bowl, stir in arrowroot, mix until dissolved, then whisk thickened mixture into soup. Simmer for 1-2 minutes, reduce heat to medium-low, then whisk in creme fraiche. Season with salt and pepper, stir in crayfish meat, and simmer until heated through, about 2 minutes. Serve warm, garnished with chives.