In large Dutch oven or stock pot, simmer smoked ham hocks in chicken broth (see note 1) for 1 hour.
Add 1/2 pound split peas and simmer 40 minutes. Add remaining split peas and simmer an additional 40 minutes or until all peas are soft. Stir frequently to keep the solids from burning on the bottom on the pan.
Remove ham hocks and cool until able to handle. Pull meat from ham hocks and shred. Add shredded ham back to soup.
During last 40 minutes of simmering, sauté onion, celery and carrot in butter until onions are translucent. Add garlic and oregano (see note 2) and sauté a little while longer until fragrant.
With 20 minutes left of simmering, add sautéed vegetables and potatoes to soup.
Add additional chicken stock if necessary to acheive desired consistency. Season with salt (see note 3) and pepper to taste. Serve hot.