In a large saucepan, combine the chicken broth, carrots, celery, potatoes and onion. Bring the mixture to a boil, cover and simmer for 10 minutes, until the vegetables are starting to get tender.
Add the chopped broccoli. Cover and simmer for another 10 minutes. The pan is going to be very full of vegetables and it might seem like it isn't "soupy" enough since the vegetables overpower the broth but I promise it will all work out when you add the cheese mixture.
In a separate medium saucepan, saute the bacon until golden and crisp. Remove the bacon to a paper towel-lined plate. Drain the grease. If you prefer you can use a combination of bacon grease and butter to prepare the cream sauce.
Melt the butter (in the same saucepan you cooked the bacon in) over medium heat. Whisk in the flour and cook for 1-2 minutes until golden, stirring constantly. Gradually whisk in the milk and cook until the mixture is bubbling and is slightly thickened, 5-7 minutes.
Stir in the cheddar cheese one handful at a time, adding another handful after the cheese that has been added has melted. Stir in the salt, pepper and dry mustard.
Slowly stir the cheese sauce into the hot broth and vegetables, whisking to combine well. Stir in the reserved bacon and add additional salt and pepper to taste, if needed. If you want the soup a bit thinner, stir in a little extra broth or milk a bit at a time until you get the consistency you are looking for. We like thick soup, so I didn't add any additional liquid.
Adapted from melskitchencafe.com