Saute onions in 2 tablespoons of butter until onions is caramelized and looks translucent.
Add minced garlic to onions and saute 1 minute max while stirring constantly. Remove from heat and set aside.
In a large soup pot with weighted bottom (if possible so soup doesn't stick) add the other 4 tablespoons of butter, flour, and cook on medium heat for approx. 5 minutes while whisking constantly with wire whip until mixture is thickened. This is an important step. If this mixture doesn't thinking properly, soup will not thicken later in the process.
Slowly add the chicken stock/broth while whisking constantly.
Slowly add Half n Half while whisking constantly. Simmer on low temp for about 20 minutes.
Then add broccolli, carrots, and the onions and garlic mixture you set aside earlier. Also add salt, pepper, and cayenne. Simmer for another 20-25 minutes. By this time, soup should have thickened some. Whisk often to break up skim that forms on top of soup.
Go ahead and stir in most, but not all of the cheese saving some for garnish. Give cheese a chance to melt. (About 1 minute DO NOT use pre-grated cheese! The flavor is just not there and that's crucial to this soup.
At this point, go ahead and serve soup into individual bowls and add a little cheese to the top of each bowl. If you have kids, add a few goldfish crackers to peak their interest.
Leftover soup can be kept in air tight containers for 1 week only! Warm in microwave. Enjoy!
