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Melt the butter in a large pot or Dutch oven over medium heat. Add the chopped onion and garlic, sautéing until the onion becomes translucent and fragrant, about 3-4 minutes. Stir in the chopped celery and carrots, continuing to cook for another 5 minutes until the vegetables begin to soften.
Add the diced potatoes and cooked ham to the pot, stirring to combine with the sautéed vegetables.
Make the Broth:
Simmer the Soup:
Cover the pot and let the soup simmer for 15-20 minutes, or until the potatoes are tender and easily pierced with a fork.
Stir in the milk and heavy cream, and season with salt, black pepper, and thyme. Continue to cook for another 5-10 minutes, allowing the flavors to meld together and the soup to thicken slightly.
If the soup is too thick, add a bit more chicken broth or milk until the desired consistency is reached. If it’s too thin, let it simmer uncovered for a few more minutes to reduce and thicken.
Serve hot, garnished with chopped green onions and shredded cheddar cheese if desired. Enjoy your creamy, comforting ham and potato soup!
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