Place the water, split peas, fennel, and celery in a large saucepan.
Bring to a boil, reduce heat, cover, and simmer for 45 minutes.
While the soup simmers, saute the shallots in the stock in a small skillet until they are translucent.
If using dried basil, stir into the shallots, then remove from heat.
Stir the shallots into the soup as it simmers.
After it has simmered for 1 hour, puree the soup in batches in a blender.
Return the blended soup to the pot.
Add the potatoes and simmer for 15 minutes.
Then add the peas and basil (if using fresh) and simmer until the peas are tender.
Stir in the parsley and remove from the heat.
Garnish with fennel sprigs, if desired.