Rinse the ham hocks (ham bone) and split peas. Combine them in a deep 7- to 8-quart stockpot with 15 cups water. Bring to a boil, skimming off any foam, and boil for 5 minutes.
Lower the heat to a simmer, add the leek, thyme, and bay leaf, and simmer gently until the peas are falling apart, about 1-1/2 hours.
Meanwhile, heat the olive oil in a 12-inch nonstick skillet over medium-high heat. Add the onions and 1/2 tsp. salt. Cook, stirring with a wooden spatula, until browned around the edges, about 5 minutes. Turn the heat down to low, cover, and cook, stirring occasionally, until the onions are tender, about 15 minutes. Stir in the carrots and celery, cover, and continue cooking, stirring occasionally, until the carrots and celery are just tender, another 15 minutes.
Transfer the ham hocks to a plate. Add the vegetables to the soup, scraping the skillet with the wooden spatula to release any browned bits. Stir in 1 tsp. salt, 1/2 tsp. pepper, the cloves, and the cayenne, and continue to simmer for 20 minutes to meld the flavors. Turn the heat off, discard the bay leaf, add the parsley, and puree the soup with an immersion blender (or in a regular blender, working in batches) until smooth.
Cut the skin from the ham hocks, and then remove the meat from the bone, discarding any large pieces of fat. Dice the meat and add it to the soup. Reheat the soup over medium-low heat, simmering and stirring occasionally, until it’s the consistency of heavy cream and pours easily from a ladle, about 10 minutes. Thin with a little water, if necessary. Season to taste with additional salt, pepper, and cayenne. Serve hot.