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Wash broccoli and trim off florets.
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In a large pot, melt butter and cook onions over medium-low heat. Onions should be softened, but not browned.
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Add broccoli florets to pan and saute until bright green, about 4 minutes.
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Add chicken broth to pot and increase heat to medium high.
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Simmer just until broccoli is tender, about 7 minutes.
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Use immersion blender, or regular blender to puree soup until smooth.
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In a regular blender, puree in two batch and take care to vent blender while pureeing to avoid burns.
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Return blended soup to pot and stir in half and half.
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Add cheeses and stir until melted.
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Season with salt and pepper.