Sprinkle chicken with 1 teaspoon thyme, salt and pepper. Place in a 6-quart or larger slow cooker and add broth, mushrooms, carrot, celery, onion, rice, sherry and garlic. Stir to combine.
Cover and cook on High for 3 hours or on Low for 6 hours.
Remove the chicken to a clean cutting board and shred. Stir the chicken back into the stew along with kale, 1/2 cup sour cream, lemon juice and the remaining 1/2 teaspoon thyme. Serve topped with a dollop of the remaining sour cream.
To make ahead: Refrigerate for up to 2 days.
Equipment: 6-qt. or larger slow cooker