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Saute onion in butter. Set aside.
Cook melted butter and flour using a whisk over medium heat for about 4 minutes. Be sure to stir frequently.
Slowly add the half-and-half, continue stirring.
Add the chicken stock whisking all the time.
Simmer for 20 minutes.
Add the broccoli, carrots and onions. Cook over low heat until the veggies are tender, about 20 minutes.
Add salt and pepper.
By now the soup should be thickened.
Pour in batches into blender and puree.
Return the puree to the pot and place over low heat, add the grated cheese; stir until well blended.
Stir in the nutmeg.
Serve.
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