Melt the butter in a heavy bottom Dutch oven. Add the red pepper flakes, shallots and carrot and sauté until soft. Season with salt and pepper.
Add the flour on top and whisk to combine. Add the broth and milk and season with salt and pepper. Turn heat to medium high and bring to a simmer. Once simmering, reduce heat to medium and let simmer for 15 minutes.
Add the broccoli florets and continue to simmer for 20 minutes until soft. Remove the pot from the heat and add the freshly shredded cheese in batches and stir until smooth. Season with salt and pepper.
Using an immersion blender, blend the soup until either completely smooth or half creamy and half chunky. I prefer to leave some of the broccoli florets in large bites. Top with extra shredded cheese and serve with the puff pastry straws.
To make the cheese straws:
Preheat the oven to 375 degrees F.
Roll out the sheet of puff pastry on a lightly floured board until it's 10 by 12 inches. Brush the surface of the pastry with the olive oil. Sprinkle each sheet evenly with the cheese and season with salt, and some pepper.
Cut the sheet crosswise with a knife 11 or 12 strips. Twist each strip and lay on baking sheets lined with parchment paper.
Bake for 10 to 15 minutes, until lightly browned and puffed.