In a large pot or Dutch oven, melt butter over medium heat until foaming subsides. Add onion, carrot, celery, ¼ teaspoon salt and ½ teaspoon pepper. Cook until vegetables are softened, 5-8 minutes. Add garlic and cook for 1 minute, until aromatic. Stir in split peas.

Add ham bone, bay leaf, 1 teaspoon fresh thyme, and cayenne pepper. Stir in chicken stock and water. Bring to a boil, reduce heat, and simmer uncovered for 60-90 minutes, stirring occasionally, until split peas are cooked down and soup is thickened to desired consistency. (Stir more frequently as the soup begins to thicken and add a little additional stock or water if it gets too thick for your taste.)

Add diced ham during the last 15 minutes of cooking. When ready to serve, remove and discard ham bone and bay leaf, then stir in remaining 1 teaspoon fresh thyme and balsamic vinegar. Season to taste with salt, if needed. Serve hot with croutons and cracked black pepper.