Bring 3 quts water, ham, and bay leaves to boil in covered large soup pot. Reduce heat and simmer until meat is tender and pulls away from bone, 2 to 2 1/2 hr. Remove ham meat and bone from broth; add split peas, thyme, and 2 potatoes and simmer until peas are tender but not dissolved, about 45 min. When ham is cool enough shred meat into bite-sized pieces and set aside. Discard rind and bone.
While peas are simmering, heat oil in large skillet until shimmering. Add onion, carrots, celery. Saute stirring frequently, until most of the liquid evaporates and vegetables begin to brown, 5-6 min. Reduce heat to medium low, add butter, garlic, and sugar. Cook vegetables, stirring frequently, until deeply browned, 20-25 min; set aside.
Add sauteed vegetables, 3 potatoes, and shredded ham to soup; simmer until potatoes are tender and peas dissolve and thicken soup to consistency of light cream, about 20 min more. Season with ground black pepper. Ladle soup into bowls, sprinkle with red onion, if using, and serve passing balsamic vinegar separately.