Rinse the peas and pick over to remove any stones or debris. Set peas aside.
Heat a Dutch oven over med-high heat. When hot, add the oil, then add the onion and bay leaves. Cook, stirring occasionally, until the onion starts to soften.
Add the thyme, celery and carrots. Cook, stirring occasionally, until the onion and celery are soft, and the carrots have started to soften.
Add the garlic and pepper. Stir. Cook about 30 seconds, stirring frequently.
Pour in the broth and peas, and stir. Bring to a simmer.
Carefully place the ham bone in the pot. Add any ham meat as well, if you have some.
Simmer on low heat for about an hour, partially covered. Stir occasionally.
Cook until peas are soft and no longer hold their shape. Taste and add salt, if needed. Remove and discard bay leaves.
Serve with some crusty bread.