1. Make chicken: Put first five ingredients in slow cooker. Pour in 6 cups cold water. Cover and cook on high until chicken is tender, about 3 1/2 hours. Remove chicken from cooker and set aside. When cool, remove meat and shred into large pieces.
2. Strain cooking liquid and set aside. Melt butter in a saucepan over medium-high heat. Add flour and whisk until smooth. Add cooking liquid to pan, increase heat to high, bring to a boil and whisk until smooth and slightly thickened. Return sauce and chicken to slow cooker. Season with salt and pepper.
3. Make dumplings: In a small bowl, combine flour, baking powder, salt and herbs. Add butter and use fingertips or a pastry cutter to work butter into dry ingredients until mixture resembles coarse crumbs. Whisk egg and milk to blend, then stir into flour mixture to form a sticky batter. Drop batter by tablespoon into cooker. Cover and cook on high until dumplings are fluffy, about 1 hour.