In a large Dutch oven or saucepan saute the onion and garlic in oil until soft (about 5-6 minutes).
Add in broth, dried split peas, water, ham bone and bay leaf; bring to a boil, reduce heat to medium-low, and simmer for about 2 hours, or until the soup thickens (add in the diced carrots the last hour of cooking).
Add in pepper to taste.
Remove the ham bone and bay leaf.
At this point the soup can be pureed if desired.
When the ham is cool, pull the meat from the bone and shred it into bite-size pieces, then return to the saucepan, or serve on the side.
Add in rice and stir until heated through.
Ladle into serving bowls, and drizzle with whipped cream if desired.