Add chicken, broth, onion, and celery to a large pot. Bring to a gentle simmer and cook until the chicken is tender, about 20 to 25 minutes. Skim foam if needed.

Remove chicken to a plate and shred into bite-size pieces. Discard onion and celery. Stir butter into the broth and season with salt and pepper. Keep the broth at a low simmer.

In a bowl, whisk flour and salt. Cut in cold butter or shortening with your fingers or a fork until it looks like small pebbles. Stir in milk until a soft dough forms. It should be moist but not sticky.

On a floured surface, roll the dough to about 1/8 inch thickness. Cut into 1 by 2 inch strips or small squares. Dust with a little flour so they do not stick.

Bring broth to a gentle simmer, not a hard boil. Drop in dumplings one at a time, stirring softly so they do not clump. Simmer 12 to 15 minutes, stirring occasionally. The broth will thicken slightly as the flour releases.

Stir in shredded chicken. Add a splash of cream if you like it richer. Taste and adjust salt and pepper. Let everything simmer 3 to 5 minutes so the flavors come together.

Ladle into bowls and enjoy that soft dumpling texture with savory chicken in a silky broth.