Rinse the split peas in a strainer, and then combine them with the stock and water in a large soup pot. Bring to a boil.
Add the ham bone, celery, 1 tablespoon of the parsley, and the tarragon. Reduce the heat to medium-low and simmer, partially covered, stirring occasionally, for 45 minutes.
Melt the butter in a saucepan over medium-low heat. Add the