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Place chicken, wild rice, carrots, celery, onion, garlic, chicken broth, and thyme in a slow cooker. Season with salt and pepper.
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Cover and cook on LOW for 6-7 hours or HIGH for 3-4 hours, until chicken is cooked through and rice is tender.
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Remove chicken, shred with two forks, and return to the slow cooker.
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Stir in heavy cream. Cook on HIGH for 15 more minutes until heated through.
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Taste and adjust seasoning. Serve hot with crusty bread.