Trim chicken of excess skin and fat.
Place chicken in a slow cooker. Pour in cold water to cover by 1 inch and add carrots, onion, celery, and bay leaf. Cover and cook 3-4 hours on high or 6-7 hours on low, or until chicken is falling off the bones.
Remove chicken from the slow cooker and set aside to cool. When chicken is cool enough to handle, remove the meat and shred into large pieces.
Strain broth into a large saucepan. Bring to a boil, reduce heat to medium-high, and cook until reduced to 6 cups, about 10 minutes.
In a large saucepan over medium-high heat, melt the butter. Add flour and whisk until smooth. Pour in the reduced broth, raise the heat to high, and bring to a boil. Cook, whisking constantly, until smooth and thickened, 8-10 minutes. Add chicken and season with salt and pepper. Transfer to slow cooker and set temperature to high.
For Dumplings: In a bowl, stir together flour, baking powder, and 1/2 teaspoon salt. Add butter and work into the dry ingredients with fingers or pastry cutter until the texture of small peas. Gently stir in the milk and egg to form a sticky batter. Drop heaping spoonfuls of dough into simmering sauce and chicken, reduce heat to low, cover, and cook until dumplings are fluffy, about 30 minutes.
You may also cook in a large Dutch oven. Cook chicken 2 hours; follow other directions but cook only 15 minutes after adding dumplings.