Cook melted butter and flour using a whisk over medium heat for 3-5 minutes. Stir constantly and slowly add the half-and-half (this is called making a roux). Add the chicken broth whisking all the time. Simmer for 20 minutes.
Add the broccoli, carrots and onions. Cook over low heat till veggies are tender for 20-25 minutes. Add salt and pepper. The soup should be thickened by now.
Pour in batches into osterizer (blender) and puree.
Return to pot over low heat and add the grated cheese; stir until well blended. Stir in the nutmeg.