Roast the Broccoli:

1. Preheat the Oven: Set the oven to 425°F (220°C).
2. Prepare the Broccoli: Toss the broccoli florets with olive oil, salt, and pepper. Spread them in a single layer on a baking sheet.
3. Roast the Broccoli: Roast for 15-20 minutes, turning halfway through, until the edges are golden brown and slightly crispy.

Prepare the Soup Base:

4. Sauté Onion and Garlic: In a large pot, heat butter over medium heat. Add the diced onion and cook for 3-4 minutes until softened. Stir in the minced garlic and cook for another 30 seconds.
5. Make the Roux: Sprinkle the flour over the onion mixture, stirring constantly, and cook for 1-2 minutes to remove the raw flour taste.
6. Add the Broth: Slowly whisk in the vegetable broth, ensuring there are no lumps. Bring the mixture to a simmer.

Blend and Simmer:

7. Add the Broccoli: Stir in the roasted broccoli, reserving a few florets for garnish if desired.
8. Blend the Soup: Use an immersion blender to puree the soup to your desired consistency. For a chunkier texture, blend only part of the soup.
9. Incorporate Dairy: Stir in the milk, heavy cream, and smoked paprika. Simmer gently for 5 minutes, being careful not to let the soup boil.

Add the Cheese and Serve:

10. Melt the Cheese: Gradually add the shredded cheddar cheese, stirring constantly until fully melted. Season with salt and pepper to taste.
11. Garnish and Serve: Ladle the soup into bowls and garnish with reserved roasted broccoli, croutons, or a drizzle of cream.