Saute onion in butter. When onions are golden, stir in garlic and cook for 3 more minutes. Set aside.
In another pan melt the 1/2 cup of butter and stir the butter and flour (make a roux) using a whisk over med. heat for 3-5 minutes. Stir constantly and slowly add the half-and-half. Add the chicken stock 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.
Pour in batches into osterizer (blender) and puree.
Return to pot and add the grated cheese and stir till well blended.