Melt butter in large stockpot over medium heat. Add onion, carrots, garlic, ham, and sugar; cover and cook until vegetables are soft, 8 to 10 minutes. Add peas, broth, water, and bay leaf; increase heat to high and bring to boil. Reduce heat to medium-low and simmer until peas are soft, about 40 minutes. Discard bay leaf and add pepper to taste. Serve.
Split-pea soup can be refrigerated for several days or frozen for a month. Because split peas continue to soak up liquid over time, the soup may be dense when reheated. If the soup is too thick, stir in 1/4 cup water at a time until the consistency is to your liking. Simmer over medium-low heat until the soup is hot.
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