In your soup pot, saute onion and garlic in the 2 T butter. Remove the veggies and set aside.

In the same pot, melt the 1/2 c butter, then add flour, and cook, whisking constantly, over medium heat for 5 minutes. (In other words, make a roux).

Stirring or whisking constantly, slowly add the half & half.

Add the chicken broth and mix well.

Bring it all to a boil, then simmer for 5 minutes.

Add the broccoli, carrots and onions. Cook over low heat 20 minutes.

Add salt and pepper to taste.

At this point, you can puree the soup, or mash it a bit with a handheld potato masher, or leave it as-is.

Return to heat and add cheese till melted.

Add cayenne.

Serve, and let people add Tabasco if they like.