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1
Place ham, bay leaves, and 3 quarts water in large stockpot or Dutch oven.
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2
Cover and bring to a boil over medium-high heat.
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3
Reduce heat to low and simmer until meat is tender and pulls away from bone, 2 to 2 1/2 hours.
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4
Remove ham meat and bone from pot and set aside.
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5
Add split peas and thyme to stock.
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6
Bring back to boil, reduce heat, and simmer, uncovered until peas are tender but not dissolved, about 45 minutes.
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7
Meanwhile, when ham is cool enough to handle, shred meat into bite-sized peas and set aside. Discard rind and bone.
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8
While peas are simmering, heat oil in a large skillet over high heat until shimmering.
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9
Add onions, carrots, and celery and saute, stirring frequently, until most of liquid evaporates and vegetables begin to brown (5 to 6 minutes).
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10
Reduce heat to medium-low and add butter, garlic, and sugar.
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11
Cook vegetables, stirring frequently, until deeply browned, 30 to 35 minutes and set aside.
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12
Add sauteed vegetables, potatoes, and shredded ham to pot with split peas.
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13
Simmer until potatoes and peas dissolve and thicken soup to consistency of light cream, about 20 minutes more.
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14
Season with pepper to taste.
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15
Ladle soup into bowls, sprinkle with red onion, if using, and serve, passing balsamic vinegar separately.
