Sauté aromatics – In a large pot, melt butter. Add onion, carrots, celery, and garlic. Cook for 5 minutes until softened.
Add peas and ham – Stir in split peas, potatoes, ham bone (or diced ham), broth, thyme, bay leaves, salt, and pepper.
Simmer – Bring to a boil, then reduce heat and simmer uncovered for 60–75 minutes, stirring occasionally, until peas break down and soup thickens.
Finish – Remove bay leaves and ham bone. Shred any meat from the bone and stir it back in.
Serve – Ladle into bowls, garnish with parsley or thyme, and serve with crusty bread.