Bring the soup mixture to a slow simmer and cover, leaving the lid slightly askew
Heat the vegetable oil in a large pot over medium heat.
Add the onion and celery and sauté until soft,4 to 5 minutes.
Add the carrots, hambone, diced ham, split peas, chicken broth, 2 cups of water and bay leaf to the pot.
Cook, stirring occasionally, until the peas have broken down completely and the soup has thickened, 50 to 60 minutes.
Remove and discard the ham bone and bay leaf.
Stir the soup to blend thoroughly and if necessary, add more water until the thickness of the soup is to your liking.
Season to taste with salt and pepper and ladle into bowls for serving.