Trim the skin from the ham hock and discard. Set the ham hock aside.
In large Dutch oven or stock pot, heat the olive oil over medium heat. Add the carrots, celery, and onion. Cook, stirring occasionally, until the onion is soft and translucent, 5-10 minutes
Stir in the split peas, bay leaves, savory, thyme; cook, stirring, for 2 minutes.
Add the ham hock and the water . Bring to a boil; reduce heat, cover, and simmer, stirring occasionally, until peas are soft, and soup is thickened, 1-1/2 to 2 hours. Add additional water if soup becomes too thick.
Transfer the ham hock to a plate and let cool slightly.
Remove the bay leaves from the soup and discard. Note: For a thicker soup use an immersion blender and give the soup a quick blend, just enough to thicken, but not fully puree the soup. Alternatively remove a cup or so of the soup, blend, and return to the pot.
Once the ham hock is cool enough to handle, remove any meat on the hock and add back to soup. Discard the fat and bone from the hock. Stir to combine the meat with the soup. Taste and generously season with salt and pepper, to taste.