In a heavy stock pot or Dutch oven on medium to medium high heat add butter, broccoli stems, onion, carrots,celery and salt and pepper and saute for 8 to 10 minutes without browning.
Sprinkle the flour in and stir to combine, about 1 minute.
Add the vegetable broth, stir to combine with the flour and bring to a slow boil.
Reduce the heat and cover the pot 10 minutes.
Use an immersion blender to puree the soup, (optional)
Add the remaining broccoli florets, diced potatoes, mustard and baking soda and simmer covered for 15 to 20 minutes to soften the florets.
Keeping the heat low, add the half and half and the grated cheese and stir in until melted.
Serve hot and garnish with more grated cheese and slivered carrot if desired.
Croutons may be a good addition too as well as pumpernickel bread.