Prep Onions and Celery: Melt 1 tablespoon butter in a skillet over medium-high heat. Saute onion and celery in hot butter until translucent, 5 - 10 minutes. Label and Refrigerate.
Prep remaining vegetables: Blanch broccoli & carrots in boiling water; Shock, Label and Refrigerate.
Prep roux: mis en place remaining ingredients.
Day 2:
Make Roux: Melt butter in a large saucepan over medium-low heat; whisk in flour; cook until flour loses its granular texture, adding 1 to 2 tablespoons of milk if necessary to keep the flour from getting too thick to stir or from burning, 3 to 4 minutes.
Gradually pour milk into flour mixture while whisking constantly.
Stir stock into milk mixture.
Bring to a simmer;
Cook until flour taste is gone and mixture is thickened, about 20 minutes.
Add onions, celery, cayenne, paprika, and dry mustard, stirring well.
Bring soup to a simmer, stirring CONStANTLY to avoid scorching the bottom of the pot.
Add parboiled vegetables to soup. Simmer an additional 5-10 minutes until broccoli is hot.
Remove from heat. Stir in Cheddar cheese, one handful at a time until cheese melts.
Season with salt and pepper to taste; Garnish with additional cheese.