Put the ham, bay leaves, and 3 quarts water in a large pot, cover and bring to a boil. Reduce the heat to low and simmer until the meat is tender and the ham pulls away from the bone (2 to 2 1/2 hours). Remove the ham and bone from the pot, and shred the ham into small pieces after it has cooled enough to handle. Discard any fat and the bone.

While the ham is simmering, heat the oil in a large skillet, and add the onions, carrots, and celery. Saute' until the vegetables begin to brown, reduce heat, add the butter, and sugar. Cook the vegetables, stirring frequently, until they are browned (30 to 35 minutes).

Add the split peas and thyme to the water in the pot. Bring back to a boil, reduce heat, and simmer (uncovered), until the peas are tender (about 45 minutes).

Add the sautéed vegetables, potatoes, and shredded ham to the pot with the split peas. Simmer until the potatoes are tender and the peas begin to dissolve and thicken the soup. Season with ground black pepper to taste.

If a smoother consistency soup is desired, immersion blend for a few minutes to produce a creamier consistency (the picture of the finished soup above has been lightly immersion blended).