Make a bouquet garni by placing basil & thyme leaves with the bay leaf in a sachet or cheesecloth. Tie the bag shut or if using cheesecloth, use kitchen twine and set aside.
In a Dutch oven over medium heat, add butter. When melted, add onion, carrot and celery; season with freshly ground black pepper. Cook until vegetables are softened, about 5 to 7 minutes. Add garlic and sauté for 1 minute. If using dried basil and thyme, add them along with the garlic.
Add split peas, ham, chicken & vegetable broth, lemon juice, and the bouquet garni; stir well and bring to a boil. Reduce heat to medium and simmer for 1 ½ hours, flipping ham every 15 minutes and stirring occasionally.
Half way through cooking, reduce the heat to medium-low and stir often. Carefully remove ham and place it on a cutting board. Remove the skin, fat and bone; discard them. Cut meat into small pieces and return to the soup. Cook for the remaining time.
Ten minutes before the end of cooking, taste and adjust if necessary; it might need more pepper and salt. Just before serving, remove the sachet and discard it. Makes 6 to 8 servings.