Melt the butter and cook the onion and stems about 5 minutes.
4 tablespoons butter, 1 medium onion, Stems from the broccoli head
Add the garlic for 1 minute, then the flour for 1-2 minutes, stirring a bit, to cook off the raw flour. This is what makes the soup nice and thick.
3-4 cloves garlic, 1/3 cup flour
Whisk in the broth gradually, then the half-and-half.
2 cups low-sodium chicken broth, 2 cups half-and-half
Add the florets, carrot, Dijon, paprika, salt, and pepper. Simmer 15-20 minutes until the broccoli is tender and the soup has thickened.
Florets from 1 large head of broccoli, 1 cup shredded carrot, 1/2 teaspoon Dijon, 1 teaspoon smoked paprika, 1 teaspoon salt, 1/2 teaspoon black pepper
Optionally blend 1-2 cups and stir it back in.
Turn off the heat and add the cheese in handfuls. Finish with hot sauce to taste.
8 ounces sharp cheddar, A few dashes of hot sauce