In a large Dutch oven over medium-high heat, cook onion in butter until caramelized.
Stir in 1/3 cup flour. Add in chicken broth, potatoes, carrots, ham and pepper; simmer until vegetables are mostly tender.
In a medium bowl, beat eggs; add salt and enough flour to make mixture a noodle consistency. (Hint: I wear rubber gloves to make dumplings.)
Tear off chunks of dough, and drop in soup. Cook for 15 minutes.
Add heavy whipping cream and enough milk to make soup your preferred consistency.
Note: Instead of ham, raw shrimp can be used. The recipe originally came from my girlfriend in western Kansas many years ago. We would make it for the hungry pheasant hunters that had been walking the fields all day since sunrise.