In a stock pot or dutch oven on medium-low heat, saute onion and garlic in 1 T butter until soft and translucent
Add 4 T butter and 4 T flour and cook until lightly browned but not dark
Whisk in 2 c. chicken broth and cook until thickened about 3-4 min.
Add 2 more c. chicken broth and cook until thickened about 5 min.
Add 2 c. ½ & ½ and cook until thickened about 5 min.
Add broccoli and carrots and simmer 15-20 minutes stirring occasionally until vegetables are tender
Whisk in salt, pepper, paprika, dry mustard and cayenne
Simmer another 15 -20 min until reduced and thickened
Turn off heat and add the grated cheese, stir until melted
Save some remaining cheese for garnish
