Add 2 tablespoons of butter to a skillet and saute the onions until just becoming translucent. Add the garlic and stir another minute, making sure not to burn it. Remove from heat and set pan aside.
In a large dutch oven or pot, melt 8 tablespoons of butter. Once melted, whisk in flour and cook for about 3-5 minutes, making sure to which constantly.
Slowly add the chicken broth, whisking constantly. Slowly add the half and half, whisking constantly.
Once all liquids have been added, allow the mixture to simmer over low heat for 15 minutes, or until it has thickened. Make sure to whisk once in awhile to break up the skin that forms on top of the liquid.
Meanwhile, chop the broccoli florets into small, bite-sized pieces, and chop the carrots. After the liquid mixture has finished simmering, add the broccoli, carrots, and onion/garlic mixture from earlier. Add all of the spices: salt, pepper, dry mustard powder, paprika, and cayenne.
Allow soup to simmer for about 20 minutes, whisking intermittently,, until it has reduced and thickened. After the soup has simmered, add in the cheese and mix until it has completely melted.
Serve in bowls and enjoy with a slice of French bread!
