Saute the onion in 2 tbsp melted butter. Set aside.
Using a wire whisk, combine 8 tbsp melted butter and the flour in a large pot over medium heat. Cook, stirring frequently, for about 4 minutes.
Slowly add the half and half; continue stirring. Add the chicken broth, whisking all the time. Simmer for 20 minutes.
Add the broccoli, carrots, and sauteed onion. Cook over low heat until the veggies are tender, about 20 minutes.
Add salt and pepper to taste.
By now the soup should be thickened. Pour in batches into a blendr an dpuree.
Return the puree to the pot and place over low heat. Add the grated cheese dnl stir unti well blended. Stir in nutmeg.
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