Place a large pot (5+ quarts) over medium heat and add 2 tablespoons olive oil.
Once the oil is hot, add 2 medium yellow onions, 3 medium carrots, and 2 ribs celery and cook for about 5 minutes, stirring often. Then add 3 cloves garlic, 1 tablespoon dried thyme, and 1 teaspoon black pepper. Stir and cook for 1 more minute.
Add 1 pound dried split peas, 8 cups water, and 2 cups vegetable broth. Bring to a boil. Then lower the heat to medium-low to keep the soup at a slow boil and cover the pot. Cook for about 90 minutes until the split peas are very tender and beginning to fall apart.
If desired, use an immersion blender to puree the peas into the broth.* You can blend until smooth or leave some peas whole in the soup.
Add 4 cups leftover ham to the soup along with 1 teaspoon salt. Simmer until the ham is warmed through. Check the seasoning and add more salt to taste as needed.
Ladel into bowls and serve warm with a sprig of fresh thyme for garnish, if desired
Notes
*If you don't have an immersion blender, you can puree the soup in a blender, working in batches. Ladle some of the soup in (do not overfill), cover, and puree. Pour into a large bowl or another pot until all the soup has been processed.Ham saltiness can vary, so start light on salt and adjust at the end.The soup will thicken as it cools. Stir in extra broth or water when reheating.If you do not add ham, this recipe will yield about 10 cups instead of 14 cups.