Place the butter in a large soup pot over medium-low heat. Stir in onion, celery, and sliced garlic. Cook slowly until the onions are translucent but not brown, 5 to 8 minutes.
Mix in ham, bay leaf, and split peas. Pour in chicken stock and water. Stir to combine, and simmer slowly until the peas are tender and the soup is thick, about 1 hour and 15 minutes. Stir occasionally. Season with salt and black pepper to serve.