Melt 1 tablespoon of butter in a skillet over medium heat. Add the diced onion and sauté until soft and translucent, about 5-7 minutes. Stir in the minced garlic and cook for another 30 seconds. Transfer to a plate and set aside.

In a large pot, melt the remaining 4 tablespoons of butter over medium heat. Add the all-purpose flour, whisking continuously for 3-5 minutes until smooth and golden, forming a roux.

Gradually whisk in the vegetable stock into the roux until smooth. Then, add the half and half while continuing to whisk. Bring to a gentle simmer and stir occasionally.

Reduce the heat to low, cover the pot, and let the soup base simmer for 15-20 minutes while whisking occasionally.

Stir in the broccoli florets, optional broccoli stems, carrots, onion, and garlic. Season with salt, black pepper, smoked paprika, mustard powder, and cayenne. Cover and cook through.

Simmer for another 20-25 minutes until the broccoli and carrots are tender. Stir occasionally to prevent sticking.

Remove from heat and stir in most of the grated cheddar cheese, saving some for garnish. Stir until completely melted.

Ladle the warm soup into bowls and sprinkle reserved cheddar cheese on top. Serve immediately.