Chop and steam carrots first, then broccoli. While broccoli and carrots are steaming, chop onion and cook until translucent in additional butter (about 4 minutes). During last minute add 1 TBSP minced garlic. When broccoli and carrots are steamed, set aside. Set onions and garlic aside. Make a roux by melting 6 TBSP butter over low heat, gradually add 6 TBSP flour whisking constantly and careful not to brown mixture. Gradually add half and half whisking constantly. Next gradually add chicken stock continuing to whisk constantly. Allow mixture to simmer on low, whisking occasionally to incorporate skin that forms. Add onion and garlic mixture to soup. Add salt, pepper, mustard, paprika and a pinch or two of cayenne. Add cheddar cheese slowly making sure it melts as you add. Next stir in steamed broccoli and carrots. Allow to simmer until all ingredients are heated through. Garnish with cheese and enjoy!