Add chicken, vegetables, and soup mixture to slow cooker crock: Place chicken in a 6-quart slow cooker. Top with onion, celery, and carrots. In a large bowl, combine chicken stock, condensed soups, 1 teaspoon salt, pepper, and thyme. Pour stock mixture over chicken and vegetables. Cover and cook until chicken is tender, 6 to 7 hours on Low or 3 to 4 hours on High.

Pull chicken out of slow cooker: Remove chicken from slow cooker with tongs; place on a cutting board and cool slightly.

Make dumpling dough: If slow cooker is on Low, increase the heat to High. Combine flour, remaining 1/2 teaspoon salt, and baking powder in a medium bowl. Add butter to flour mixture; cut in butter until mixture resembles coarse meal. Add milk and stir to form a soft dough.

Roll out and cut dumplings: Place dough on a floured surface; roll into a rectangle at 1/4-inch thickness. Cut dough lengthwise into 1-inch strips; cut dough crosswise into 1 1/2- to 2-inch pieces.

Add dumplings to soup: Drop dough pieces into soup mixture in slow cooker. Cover and cook on High until dough is chewy-tender, 20 to 30 minutes.

Shred chicken, and add to soup: Remove and discard skin and bones from chicken. Shred meat with 2 forks. When dumplings are done, stir chicken into soup. Add salt to taste, if desired. Ladle into bowls.