Heat olive oil in a heavy soup pot or Dutch oven. Add onion and garlic; sauté until soft, about 5 minutes. Stir in herbes de Provence, turmeric, bay leaves, and cayenne pepper; cook and stir until fragrant, 2 to 3 minutes.
Add vegetable stock, water, split peas, 1/2 of the potatoes, and 1/2 of the carrots. Bring to a boil, then reduce the heat to low. Cover and simmer until peas are nearly tender, about 35 minutes.
Stir in remaining potatoes and carrots. Add ham, cover, and simmer until vegetables are tender, adding more stock or water as needed, about 25 more minutes.
