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Sauté the onion and garlic in butter until softened.
Add the flour and cook for 1-2 minutes to make a roux.
Slowly whisk in the broth and milk, and bring to a simmer.
Add the broccoli and cook until tender, about 10-15 minutes.
Remove from heat and blend with an immersion blender until smooth.
Stir in the cheddar cheese until melted and combined.
Season with salt and pepper to taste.