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 1 ½ cups of chicken broth until smooth, then pour in the other 4 cups. Add the sugar, pureed 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 1/16 teaspoon of turmeric.
Garnish with cheese is optional, but does provide extra cheesiness. Serve hot in a sourdough bread bowl or with slices of sourdough bread or French baguette.