Poach the chicken: Add chicken breasts, broth, celery, carrots, and onion to a large pot. Bring to a boil, then reduce heat and simmer gently until the chicken is cooked through, about 20 minutes.

Shred the chicken: Remove the chicken from the pot and let it cool slightly before shredding with two forks. Keep the broth simmering gently on low heat.

Make the dumpling dough: In a bowl, whisk together flour, baking powder, and 1/2 teaspoon salt. Cut in 2 tablespoons butter until crumbly, then stir in milk just until a soft dough forms.

Rest the dough: Let the dough rest for 5 minutes on a lightly floured counter so it's easier to roll and won't shrink back.

Roll and cut dumplings: Roll the dough out to about 1/8 inch thick and cut into 2-inch strips or squares. Dust lightly with flour so the pieces don't stick together.

Drop dumplings into broth: Bring the broth back to a gentle simmer, not a hard boil. Drop in the dumplings one at a time, stirring gently so they don't clump.

Cook until tender: Cover the pot and simmer until the dumplings float to the top and turn tender, about 10 minutes. Avoid stirring too much so they stay intact.

Stir in the chicken: Add the shredded chicken back into the pot along with remaining butter, salt, and black pepper. Simmer uncovered for 5 more minutes to warm through.

Check the broth consistency: The broth should coat the back of a spoon lightly. If it's too thin, simmer a few more minutes uncovered; if too thick, stir in a splash of broth.

Serve warm: Ladle into bowls while hot, with extra black pepper sprinkled on top if you like it peppery like the diner version.