In small fry pan, melt 1 tablespoon butter, adding diced onion. Mix till onions on translucent. Add minced garlic, stirring frequently so it doesn't burn. 30 seconds or so. Take pan off heat.
In 6 quart stockpot or dutch oven, add 4 tablespoons butter. Once melted, add 1/4 cup flour. Whisking constantly until flour is totally incorporated. This is the roux that will thicken the soup. This takes about 3-4 minutes. Pan should be on low heat.
Add stock continuing to whisk. Then add milk whisking to blend.
When liquid is blended, let simmer for 15-20 minutes till it has reduced slightly. You will want to keep whisking periodically to remove the film that naturally forms across the top of the liquid.
Add broccoli, carrots, onion and garlic mixture. Mix. Add salt, pepper, dry mustard powder and paprika stirring to incorporate all ingredients.
Let simmer 20 minutes to cook vegetables and mingle flavors.
Shred 8 ounces or more of cheddar cheese (I used medium sharpness). Suggestion to not buy the pre-shredded cheese as it doesn't melt easily.
Turn heat off pot, add 3/4 of the shredded cheese to soup mix saving the rest for the top of each bowl. Note: Cheese lovers may want more cheese.