In a soup pot over medium heat, melt the butter. Add the chopped onion and sauté for 4 to 5 minutes until softened.
Stir in the minced garlic and cook for 30 seconds.
Add the flour and cook for 1 to 2 minutes, stirring frequently. This step is important to eliminate the raw flour taste.
Slowly whisk in the chicken or vegetable broth until the flour is fully dissolved.
Whisk in the Dijon mustard.
Stir in the cream, broccoli, and carrots. Increase the heat to high and bring the soup to a boil. Then reduce the heat so it simmers gently, with the pot’s lid slightly ajar. Medium-low heat works well to maintain a steady simmer without a vigorous boil.
Let the soup simmer for 15 to 20 minutes, or until the broccoli and carrots are tender and the soup has thickened to your preference. Stir occasionally to prevent sticking and to ensure consistent heat.
Lower the heat to low and gradually stir in the shredded cheddar cheese. Cook for a few more minutes until the cheese has melted completely.
Add the cayenne pepper if using, and season with salt and pepper to taste. Optionally, pulse the soup a few times with an immersion blender directly in the pot to blend part of it. The soup will thicken further as it cools. Add more broth if you prefer a thinner consistency.
Serve immediately, garnished with extra shredded cheddar on top of each bowl.