Place ham bone, bay leaves and three quarts water in large pan, bring to a boil, reduce heat and simmer for about 2 1/2 hours till tender.
Remove bone and when cool enough to handle, shred meat and set aside. Skim as much fat as possible from the top of the broth. (This is a good place to stop if you’re not making it for tonight’s meal – just refrigerate the broth and skim off any fat that accumulates on the top the next day.)
Add split peas and thyme, bring to a boil, reduce heat and simmer 45 minutes.
While the split peas simmer, heat oil in a large skillet and saute onions, carrots, and celery until liquid evaporates and vegetables begin to brown on the edges, five to six minutes. Reduce heat and add sugar, garlic, and butter, and cook slowly, stirring now and then till vegetables reach a deep caramelized color, about 30 to 35 minutes.
After the peas have cooked for about 45 minutes, add in the cooked vegetables, the potatoes, and the shredded ham meat. Simmer till consistency of heavy cream, about 20 more minutes. Season with pepper.
Drizzle with balsamic vinegar and minced red onion, if desired.
Split Pea Soup will thicken and gel in the fridge. Add milk or water when reheating to bring to desired consistency.