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Heat the olive oil or butter in a large pot over medium heat. Add the chopped onion and garlic, cooking for 2-3 minutes until softened and fragrant.
Add the broccoli florets and diced potato to the pot, stirring to combine. Pour in the vegetable broth (or chicken broth), and season with salt and black pepper. Bring the mixture to a boil.
Reduce the heat and let the soup simmer for about 15-20 minutes, until the broccoli and potatoes are tender.
Use an immersion blender to blend the soup directly in the pot until smooth, or carefully transfer the soup to a blender in batches to blend until creamy.
Return the soup to the pot, and stir in the shredded cheddar cheese, allowing it to melt completely. If you'd like a richer soup, add the heavy cream and stir to combine.
Taste the soup and adjust the seasoning with additional salt or pepper, if desired.
Serve hot, garnished with extra shredded cheddar cheese for a cheesy finish!
Serving and Storage Tips:
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