Mix chopped stuffing, eggs, flour and the 1/4 cup breadcrumbs in a bowl just until combined. Refrigerate for 15 minutes so the bread absorbs moisture.
Melt butter in a 6-quart soup pot and cook onion, carrots and celery for 6 minutes. Add turkey broth and pepper and simmer for 15 minutes until vegetables are tender.
Meanwhile, bring poaching water and salt to a very gentle simmer in a wide 4-quart saucepan. Add 1 teaspoon of the dumpling mixture as a test and simmer for 3 minutes to check whether it holds together, then remove and discard that small test piece. If it breaks apart, stir reserved breadcrumbs into the remaining mixture a tablespoon at a time before shaping.
Shape the adjusted mixture into 12 equal balls with damp hands. Lower them into the gently simmering water without crowding, working in 2 batches if needed, and poach for 12 to 15 minutes until firm enough to lift and centers reach 165°F.
While dumplings cook, add turkey to the soup and simmer gently for 5 to 8 minutes until it reaches 165°F. Keep the soup hot without a hard boil.
Lift dumplings with a slotted spoon, place 2 in each bowl and ladle turkey, vegetables and clear broth around them.