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Chop the broccoli stems and set them aside. Chop the broccoli florets into bite-sized pieces.
Melt the butter in a large soup pot over medium-high heat. Add the onions and garlic and cook until the onions are translucent, about 3 minutes.
Add the broccoli stems, potato, chicken stock, salt, and pepper to the pot and bring it to a boil. Reduce the heat and simmer, uncovered, for 5 minutes.
Add HALF the broccoli florets and simmer the soup for another 5 minutes, or until the potato and broccoli stems are soft. Blend the soup using an immersion blender or carefully working in batches with a regular blender.
Add the remaining broccoli and the cheddar cheese to the pot and stir well. Let the broccoli cook for 3-4 minutes to soften the broccoli.