1 smoked bone-in picnic ham (about 2 1/2 pounds)
4 bay leaves
1 pound split peas (2 1/2 cups), rinsed and picked through
1 teaspoon dried thyme
2 tablespoons olive oil
2 medium onions , chopped medium
2 medium carrots , chopped medium
2 medium stalks celery , chopped medium
1 tablespoon unsalted butter
2 medium cloves garlic , minced
pinch granulated sugar
3 small new potatoes , scrubbed and cut into medium dice
Ground black pepper
minced red onion (optional)
1. Bring 3 quarts water, ham, and bay leaves to boil, covered, over medium-high heat in large soup kettle. 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 broth; add split peas and thyme and simmer 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.
2. While ham is simmering, heat oil in large skillet over high heat until shimmering. Add onions, carrots, and celery; sauté, stirring frequently, until most of the liquid evaporates and vegetables begin to brown, 5 to 6 minutes. Reduce heat to medium low; add butter, garlic, and sugar. Cook vegetables, stirring frequently, until deeply browned, 30 to 35 minutes; set aside.
3. Add sautéed vegetables, potatoes, and shredded ham to soup; simmer until potatoes are tender and peas dissolve and thicken soup to the consistency of light cream, about 20 minutes more. Season with ground black pepper. Ladle soup into bowls, sprinkle with red onion, if using, and serve, passing balsamic vinegar separately.