In a large stockpot over medium heat, melt the butter and add the onion, carrot, and celery. Cook, stirring frequently, until the veggies start to soften, about 3-4 minutes. Add the garlic and cook for an additional 1 minute, stirring constantly.
Add the ham bone and slices, split peas, chicken broth, water, thyme, salt, pepper, and bay leaf.
Bring the soup to a boil, then reduce heat to a simmer, stir, and cover. Allow to simmer for 60-75 minutes, or until the peas are tender.
Remove the ham bone and slices and chop all the meat into small pieces.
Use an immersion blender to partially blend the soup if desired, then stir the chopped ham back into the soup. Serve immediately with fresh thyme and croutons.