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Export 12 ingredients for grocery delivery
In a large pot, melt butter over medium-high heat.
Add in onion and carrot; sauté, stirring frequently, until vegetables have softened, 5 minutes. Stir in the garlic and cook for 1 minute longer.
Season with the Italian herbs, cracked red pepper, and salt and pepper, to taste.
Add the leftover ham bone (or ham hock) to the pot, along with the chicken broth and potatoes; bring to a boil.
Turn down heat to medium and simmer for 12-15 minutes until the potatoes are tender, but not mushy.
Discard ham bone. (If using a ham hock, cut away the meat and chop into 3/4-inch cubes.)
Add in the half-n-half and diced ham; cook for another 2 minutes to warm through.
Ladle soup into bowls and serve with a little garnish of freshly chopped parsley and some oyster crackers.