Rinse the split peas under cold water and drain.
In a large pot, heat some olive oil over medium heat. Add the chopped onion, carrots, and celery. Cook until softened, about 5 minutes.
Add the minced garlic, bay leaf, and dried thyme. Cook for another minute until fragrant.
Pour in the vegetable or chicken broth and add the split peas to the pot.
If using, add the ham hock or smoked ham for extra flavor.
Bring the soup to a boil, then reduce the heat to low. Cover and simmer for about 1 to 1.5 hours, or until the split peas are tender and the soup has thickened, stirring occasionally.
Once cooked, remove the bay leaf and any large pieces of ham hock. Use an immersion blender or transfer a portion of the soup to a blender and puree until smooth, if desired.
Season with salt and pepper to taste. Serve hot, garnished with fresh herbs if desired.