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A hearty soup made with smoked ham and split peas, flavored with caramelized vegetables.
The caramelization of the vegetables is crucial for flavor.
Place ham, bay leaves, and 3 quarts water in large stockpot or Dutch oven. Cover and bring to a boil over medium-high heat. Reduce heat to low and simmer until meat is tender and pulls away from bone, 2 to 2 1/2 hours. Remove ham meat and bone from pot and set aside.
Add split peas and thyme to stock. Bring back to boil, reduce heat, and simmer, uncovered, until peas are tender but not dissolved, about 45 minutes. Meanwhile, when ham is cool enough to handle, shred meat into bite-sized pieces and set aside. Discard rind and bone.
While split peas are simmering, heat oil in large skillet over high heat until shimmering. Add onions, carrots, and celery and sauté, stirring frequently, until most of liquid evaporates and vegetables begin to brown, 5 to 6 minutes. Reduce heat to medium-low and add butter, garlic, and sugar. Cook vegetables, stirring frequently, until deeply browned, 30 to 35 minutes; set aside.
Add sautéed vegetables, potatoes, and shredded ham to pot with split peas. Simmer until potatoes are tender and peas dissolve and thicken soup to consistency of light cream, about 20 minutes more. Season with pepper to taste.
Ladle soup into bowls, sprinkle with red onion, if using, and serve, passing balsamic vinegar separately.