Heat olive oil in a large pot over medium-high heat. Add onion and celery and saute 3 minutes. Add garlic and saute 1 minute longer.
Pour in chicken broth and water. Add split peas, bay leaves and thyme. Season lightly with salt and with pepper to taste
Nestle ham into soup. Bring mixture to a boil, then reduce to low. Cover and let simmer, stirring occasionally until peas and ham are tender, about 60 minutes.
Remove ham from soup, let rest 10 minutes then shred or dice meat portion into pieces, cover.
Add carrots to soup. Cover soup and continue to simmer, stirring occasionally, about 30 minutes
Stir ham into soup, season with more salt as needed.