Melt 1 tablespoon of butter in a large pot. Add finely chopped onions, celery, and carrots and mix until evenly coated. Cook until softened - about 10 minutes. Season with salt and pepper.
Finely chop the broccoli florets and stems into small pieces and add to pot. Stir well to mix evenly and then add 2 cups of broth. Bring to a soft boil, cover and let simmer for about 20 minutes.
In a separate pot add remaining butter and melt on medium heat. Once melted add flour, mustard, salt and pepper and whisk to create a creamy roux.
Add remaining cup of broth and whisk so well blended and mixture begins to thicken.
Pour in cream while continuously whisking and blend until smooth.
Fold in the cheese and stir to blend and melt.
Pour the cream mixture into the broccoli veg mix and sir to gently mix. Continue simmering for about 15 more minutes. Remove from heat to cool slightly.
Using a hand blender puree the soup directly in the pot and serve immediately.