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In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion and garlic, and sauté until softened and fragrant, about 5 minutes.
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Sprinkle the flour over the onion and garlic mixture, stirring constantly to make a roux. Cook for 2-3 minutes to remove the raw flour taste.
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Slowly pour in the chicken or vegetable broth, stirring constantly to prevent lumps from forming. Then add the half-and-half, bay leaf, nutmeg, salt, and pepper. Bring the mixture to a simmer, stirring occasionally.
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Once the mixture is simmering, add the chopped broccoli and diced carrots. Cook for about 15-20 minutes, or until the vegetables are tender.
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Remove the bay leaf from the soup. Using an immersion blender or regular blender, blend the soup until smooth and creamy.
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Return the soup to the pot if using a regular blender. Stir in the shredded cheddar cheese until melted and well combined.
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Taste the soup and adjust seasoning if necessary. If the soup is too thick, you can add more broth or half-and-half to reach your desired consistency.
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Serve the creamy broccoli cheddar soup hot, garnished with extra shredded cheddar cheese if desired. Enjoy your homemade Panera Bread-inspired soup!