Rinse and check the peas. Pour the split peas into a colander and rinse them under cold water. Pick through them and throw away any small stones or shriveled peas.
Sauté the vegetables. Heat the olive oil in a large pot or Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 6 to 8 minutes, stirring often, until the onion softens and the carrots start to turn bright.
Add garlic and seasonings. Stir in the garlic, thyme, and bay leaf. Cook for 1 minute until the garlic smells fragrant.
Add peas, ham hock, and liquid. Pour in the rinsed split peas, then add the ham hock. Pour in the chicken broth and water. The liquid should cover everything by about 2 inches.
Simmer low and slow. Bring the soup to a boil, then reduce the heat to low. Cover with the lid slightly open and simmer for 90 minutes, stirring every 15 minutes or so. The peas will slowly break down and thicken the soup.
Remove the ham hock and shred the meat. Lift the ham hock out onto a cutting board. When it is cool enough to handle, pull the tender meat off the bone and shred it into bite-size pieces. Throw away the bone, skin, and fat.
Finish and serve. Stir the shredded ham back into the soup. Remove the bay leaf. Taste and add salt and pepper as needed. If the soup is too thick, stir in a little hot water. Ladle into bowls and top with croutons if you like.
