In a large saucepan set over medium heat, add the onions, carrots, and celery after heating the olive oil. Cook, stirring often, for 5 minutes.
Add the paprika, oregano, black pepper, tomato paste, and garlic. Cook for one to two minutes, stirring often.
Stir together the navy beans, water, chicken broth, smoked ham shank, and bay leaf in the saucepan. Turn the heat up to high, boil the liquid, cover, reduce the heat to medium-low, and simmer for 60 minutes, stirring now and again.
After taking the smoked ham shank out of the saucepan, transfer it to a chopping board. Keep the soup covered and simmer it further. Once the smoked ham shank has cooled down sufficiently, take the flesh off of the bones. Throw away the bones and any bits of fat. Return the chopped meat to the pot.
The navy bean soup should be cooked for a further fifteen minutes. If a few beans are soft to the taste, the soup is finished. Cook the beans covered if they are still a bit stiff. After the beans have reached the right consistency, check them about every fifteen minutes.
If necessary, taste the navy bean soup and add more salt. For comparison, we put ½ tsp into our soup.
Extract the bay leaf and dispose of it.