Start with Sauté: Heat the olive oil in a large pot over medium-high heat. Sauté the chopped yellow onion and celery for 3 minutes, then add the minced garlic and continue sautéing for another minute.
Combine Broth and Seasonings: Add both the chicken broth and water into the pot. Mix in the dried split peas, bay leaves, and thyme. Lightly salt and adjust the black pepper to taste, planning to reassess saltiness after cooking with the ham.
Incorporate Ham and Simmer: Add the ham bone or shanks into the soup mixture, ensuring it's fully submerged. Bring the soup to a boil before reducing the heat to a simmer. Cover the pot and let it simmer, occasionally stirring, until the peas and ham soften, about 60 to 80 minutes.
Handle the Ham: After the ham is cooked and tender, remove it from the pot and let it cool for 10 minutes. Then, shred or cut the ham into bite-sized pieces, keeping them covered.
Add Carrots and Continue Simmering: Introduce the chopped carrots to the simmering soup. Cover the pot again and keep simmering, stirring now and then, until the peas break down and the carrots are cooked through, about 30 more minutes.
Finish and Serve: Return the ham to the soup, adjusting the salt to taste. Serve the soup hot, garnished with fresh parsley if using.