In a large pot or Dutch oven, melt butter until foaming subsides. Add onion, carrot, celery, 1/4 teaspoon salt and 1/2 teaspoon pepper. Cook until vegetables are softened and just beginning to brown, 5-8 minutes. Add garlic and cook for 1 minute, until aromatic. Stir in split peas.
Add bay leaf, and 1 teaspoon oregano. Stir in chicken stock. 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 tastes.)
Add diced ham during the last 15 minutes of cooking. When ready to serve, remove and discard bay leaf. Season to taste with salt, if needed.