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, ham bone, diced ham, split peas, chicken broth, 2 cups of water and bay leaf to the pot.
Bring the soup mixture to a slow simmer and cover, leaving the lid slightly askew.
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.
Instant Pot Method
Press the "sauté" button on the Instant Pot, let it heat for about 1-1/2 minutes, then add the vegetable oil, onion and celery. Sauté until soft, 3 to 4 minutes.
Cancel the "sauté" function and add the carrots, ham bone, diced ham, split peas, chicken broth, 2 cups of water and bay leaf to the pot.
Lock the cover onto the Instant Pot and use the "manual" function to set the pot to cook for 18 minutes at high pressure.
Once the time is up, allow the pressure to release naturally for 15 minutes, then release any remaining pressure by turning the steam valve to the venting position. Be sure to keep your hands and face away from the top of the Instant Pot as you do.
Unlock the cover, remove the ham bone and bay leaf and discard.
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.
Tips for Making This Recipe
Split Pea Soup Leftovers:
Split pea soup will thicken even more once refrigerated, so it's helpful to have some additional low-sodium chicken broth on hand to thin it a bit when reheating.