Put chicken broth, chopped celery, shredded cabbage, chopped onion, and peeled chopped carrot into a large pot and bring it to a rolling boil. Lower the heat and simmer for 20 minutes, until the vegetables soften.
In a bowl, combine the half and half cream with the milk, stirring until evenly mixed.
Melt the butter over medium heat in a separate saucepan. Add the flour along with salt and pepper, mixing continuously to create a smooth roux. Slowly pour in the cream and milk blend while stirring constantly. Keep cooking until thickened to your liking.
Slowly stir the creamy mixture into the pot with the cooked vegetables, blending well. Fold in the cubed ham and sprinkle with dried thyme. Heat the whole soup through.
Optionally, top with chopped parsley for a fresh finish. Serve hot and enjoy.
