For the vegetables, in a 6-quart or larger pot, add the chicken broth, carrots, celery, potatoes and onions. Bring the mixture to a boil, reduce the heat, cover, and simmer for 10 minutes until the vegetables begin to soften. The vegetables will continue to cook in the following step.
Add the broccoli. Cover and simmer for 8 to 10 minutes. (The pot might seem overly full with veggies and not "soupy" enough, but it will thin out once the cheese sauce is added.)
In a separate medium saucepan over medium heat, saute the bacon until golden and crisp. Remove the bacon to a paper towel-lined plate. Drain the grease and return the pot to the stove.
Add the butter and melt over medium heat. Whisk in the flour and cook for 1-2 minutes until golden, stirring constantly. Slowly pour in the milk, whisking constantly, and cook until the mixture is bubbling and slightly thickened, 5-7 minutes.
Add the cheese a handful at a time, stirring until melted before adding more cheese. Whisk in the salt, pepper and dry mustard.
Pour the cheese sauce into the pot with the vegetables and stir to combine. Heat until gently bubbling and warmed through.
Stir in the reserved bacon and add additional salt and pepper to taste, if needed. Add additional milk or broth to thin the soup, if needed. The soup will continue to thicken as it cools.
Notes
Bacon: the bacon grease can be used for the base of the white sauce instead of using the butter in the recipe (just make sure you have at least 4 tablespoons, otherwise, supplement with butter). Also, because more bacon is always better, I almost always increase the bacon to a 12-ounce package (and then reserve some of the cooked bacon to top individual servings).Broth: increase the broth to four cups for a slightly less thick soup.