Melt the butter in a large heavy bottomed pot over medium high heat until it's bubbly. Add the onions and carrots and saute for 3 minutes.
Add the garlic, dried spices, and Dijon mustard and cook for 1 minute, stirring constantly with a wooden spoon.
Sprinkle in the flour and cook for 2 minutes, whisking constantly.
Pour in ½ cup of chicken broth, than scrape off the brown fond on the bottom of the pot with a wooden turner. Continue whisking in 1½ cups of chicken broth until smooth, then pour in the other 4 cups.
Add the sugar, puréed carrots (see notes), and a few dashes of Tabasco sauce. Mix it well, then add in the broccoli florets.
Cover and bring to a boil, then reduce the heat to medium low and simmer for 10 minutes to cook the broccoli.
For a thicker soup, whisk together 2 tablespoons cornstarch and 2 tablespoons of cold water until smooth. Add it to the soup and let it simmer for 2 minutes on medium low heat.
Stir in the whole milk and the cream and let it warm through for 2 minutes. Remove from the heat and gradually whisk in the shredded cheese one handful at a time. Taste and adjust seasonings, possibly adding a tiny bit more of Dijon and Tabasco. If you like your soup more yellow, add another pinch of turmeric.
Garnish with cheese (it's optional, but does provide extra cheesiness). Serve hot in a sourdough bread bowl or with slices of sourdough bread or French baguette.