Place chicken and water in a large pot over medium heat. Bring to a simmer and cook until chicken is no longer pink in the center and reaches 165 degrees F (74 degrees C), about 20 minutes. Transfer chicken to a cutting board; let cool slightly, then shred.

Return shredded chicken to the pot. Stir in butter, salt, and ground black pepper. Simmer until flavors meld, about 15 minutes.

Stir baking mix and milk together in a medium bowl until a soft dough forms.

Whisk flour and cold water together in a small bowl until smooth. Stir into the broth and bring to a boil, stirring constantly, until slightly thickened.

Drop baking mix mixture by tablespoonfuls into the boiling water, neatly around the chicken. Reduce heat to a simmer and cook, uncovered, 10 minutes. Then cover and cook 10 minutes more. (Sneak a spatula under the dumplings to prevent chicken from sticking to the bottom of the pot.) Serve hot.