In a large cast iron or soup pot add 1 tablespoon olive oil. Add onion and garlic and sautre until fragrant, about one minute.
Add celery, split peas, bay leaf, and fresh thyme. If you're using a ham bone, add this now. Using a bone adds so much flavor - I high recommend! If you only have sliced ham, that's ok! Reserve for later - don't add now.
Top with broth and water, bringing to a light simmer. Cover and cook about 1 1/2 hours, stirring sporadically.
After soup has cooked for 1 1/2 hours, add sliced carrots. If you didn't have a ham bone and are using sliced ham, now is the time to add it! If your ham wasn't particluarly smoky, I highy recommend adding a drop or two of liquid smoke. A little goes a long way - so don't over do it! Cook for an additional 20-30 minutes until carrots are soft.
Reove from heat, and taste. Adjust seasoning, though I doubt you'll need salt!