Rinse and Drain Peas: Rake through split peas to remove debris, then rinse under cold water.
Sauté Vegetables: Heat olive oil in a large pot. Add onion, carrots, and celery. Cook for 5 minutes until softened. Add garlic and sauté for 1 minute.
Add Peas and Ham: Stir in split peas, ham hock (or diced ham), bay leaves, and thyme.
Pour Liquids: Add broth and water. Bring to a boil, then reduce heat to a simmer.
Simmer: Partially cover and cook for 1–1.5 hours, stirring occasionally, until peas are tender and soup thickens.
Season: Remove bay leaves. For a smoother texture, blend lightly with an immersion blender. Adjust salt and pepper to taste.
Serve: Ladle into bowls. Garnish with fresh parsley or croutons if desired.