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Heat the olive oil or butter in a large pot over medium heat. Add the onion and sauté for about 5 minutes, until translucent. Add the garlic and cook for 1 minute.
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Add the diced potatoes and broth. Bring to a boil, then reduce the heat and simmer for 15–20 minutes, or until the potatoes are tender.
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Use an immersion blender to blend the soup until smooth. For a chunkier soup, blend only half of the potatoes and leave the rest intact.
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Stir in the heavy cream, cheddar cheese, and sour cream. Heat over low heat until the cheese has melted and the soup is heated through.
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Season with salt, pepper, and smoked paprika or cayenne, if desired.
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Ladle into bowls and top with shredded cheddar, crumbled bacon, and green onions. Serve warm.