Step 1: Rinse the hambone under cold water. Place it in a large stockpot (8-quart minimum) and cover with 10-12 cups of cold water. Add the bay leaf and whole black peppercorns.
Step 2: Bring the water to a boil over high heat, then immediately reduce the heat to low. Cover the pot and let it simmer gently for 2 to 3 hours, or until the meat is very tender and falling off the bone, and the broth is rich and fragrant.
Step 3: Carefully remove the hambone from the pot and place it on a cutting board to cool. Once cool enough to handle, pick off all the edible ham meat. Shred or dice the ham and set aside. Discard the bone, fat, and gristle. Strain the broth through a fine-mesh sieve to remove any bone fragments or peppercorns, then set aside.
Step 4: While the bone is cooling, in a separate large pot or Dutch oven, heat the olive oil or butter over medium heat. Add the chopped onion, carrots, and celery. Sauté for 5-7 minutes, stirring occasionally, until the vegetables have softened and become fragrant.
Step 5: Add the minced garlic to the pot with the vegetables and cook for another minute until fragrant. Pour in the strained hambone broth. Add the diced potatoes and dried thyme. Bring the soup to a boil, then reduce the heat to low, cover, and simmer for 15-20 minutes, or until the potatoes are tender.
Step 6: Stir in the reserved shredded ham meat during the last 5-10 minutes of cooking, just enough to heat it through. Taste the soup and season with salt and freshly ground black pepper as needed. Remember that hambones can vary in saltiness, so always taste before adding more salt.
Step 7: Ladle the hot hambone soup into bowls. Garnish with fresh chopped parsley, if desired. Serve immediately with your favorite crusty bread or cornbread.