Heat oil in a large pot over medium. Sauté onion, carrot, and celery until softened, about 5–7 minutes.
Add garlic and cook 30 seconds until fragrant. Stir in split peas, ham hock, bay leaves, thyme, and broth.
Bring to a boil, reduce heat, and simmer gently, partially covered, for 60–75 minutes until peas break down.
Remove ham hock, shred the meat, and return to pot. Discard bone and bay leaves. Season with salt and pepper.
For creamier texture, mash some peas with a potato masher or pulse with an immersion blender. Reheat if needed and serve.