Sort through and rinse the split peas. Soak them in cold water for 20–30 minutes while you prep the other ingredients. Drain using a fine mesh strainer before using.
Heat the olive oil in a large pot over medium heat. When the oil is hot but not smoking, add the onion and carrot. Cook for 3 minutes, stirring frequently.
Stir in the garlic, oregano, cumin, and black pepper. Cook for 1 minute, stirring constantly.
Add the smoked ham hocks, chicken broth, and bay leaf to the pot. Stir well. Raise the heat to medium-high. Once the broth starts to simmer, lower the heat to medium-low. Cover and simmer for 20 minutes.
Stir in the soaked and drained split peas. Raise the heat to medium-high. When the soup returns to a simmer, lower the heat again to medium-low. Cover and cook for 30 minutes, stirring occasionally to prevent sticking. Adjust the heat as needed to maintain a simmer.
Use tongs to remove the ham from the pot and set it on a cutting board to cool.
Add the diced potatoes to the soup. If you had them soaking in water, be sure to drain them first. Cover the pot and continue cooking.
Once the ham is cool enough to handle, remove and discard the fatty skin and bones. Chop the meat into small pieces and return it to the pot. Stir well.
Cover and continue cooking for 20–25 minutes, or until the potatoes are tender and the split peas are falling apart. Stir occasionally, more often as the soup thickens, to prevent sticking.
Remove and discard the bay leaf. Taste the soup and add salt if needed. (As a reference, we did not add any extra salt to ours.)
Serve hot with crusty bread or a side of white rice, if desired.