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Heat the olive oil in a Dutch oven medium heat Add the onion, stirring into the oil, and let simmer until softened. Add in the garlic and stir – cook for 1-2 minutes. Remove to a side dish I fusing a bone.
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If using a ham bone or hock, put into the Dutch oven with a little more oil and let bone on all sides. If using ham pieces, skip this step and add directly to the onions.
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Return onions to the pan, add the chicken stock and about half the water. Add in the split peas, stirring to break up any clumped together. If the water and stock do not cover everything, add more water. Add 2 tsp salt and 1 tsp pepper.
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Simmer for 1-2 hours until whatever meat on the bone is starting to fall off. (30-60 minutes is enough for leftover ham pieces).
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Remove the bone, cool until you can handle, and pick off any meat, and return the meat to the soup. Add the carrots. Adjust seasoning to taste. Simmer until carrots are tender. The soup will be quite thick, so add more water if needed to desired consistency. It will thicken upon standing.
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Serve with good crusty bread.