Mix stuffing, eggs, flour, and 1/4 cup breadcrumbs just until combined. Refrigerate 15 minutes to absorb moisture.
Melt butter in 6-quart soup pot; cook onion, carrots, and celery 6 minutes. Add broth and pepper; simmer 15 minutes until vegetables tender.
Meanwhile, very gently simmer poaching water and salt in wide 4-quart saucepan. Test 1 teaspoon mixture for 3 minutes; discard test piece. If it breaks apart, add reserved breadcrumbs to mixture by tablespoon before shaping.
With damp hands, form 12 equal balls. Lower into gently simmering water without crowding, in 2 batches if needed. Poach 12 to 15 minutes until firm enough to lift and centers 165°F.
Meanwhile, add turkey to soup; gently simmer 5 to 8 minutes to 165°F. Keep hot without hard boiling.
Slotted-spoon lift dumplings; put 2 in each bowl. Ladle turkey, vegetables, and clear broth around.