Step 1: Prepare the Garlic Croutons

1. Preheat the oven to 375°F (190°C).
2. In a large mixing bowl, combine cubed bread, olive oil, minced garlic, salt, and dried Italian herbs (if using). Toss until the bread is well-coated.
3. Spread the bread cubes in a single layer on a baking sheet.
4. Bake for 10-12 minutes, or until golden brown and crispy, turning halfway through the baking time. Set aside to cool.

Step 2: Start the Soup Base

1. In a large pot, heat the olive oil and butter over medium heat.
2. Add the finely chopped onion and cook until translucent, about 5-6 minutes.
3. Add the minced garlic and sauté for another minute until fragrant.

Step 3: Cook the Vegetables

1. Stir in the diced carrot and cook for about 4 minutes.
2. Add the broccoli florets and continue to cook for another 5 minutes, stirring occasionally.
3. Sprinkle the flour over the vegetables, stirring well to coat everything evenly. Cook for 2 minutes to remove the raw flour taste.

Step 4: Add Broth and Simmer

1. Pour in the chicken or vegetable broth, stirring constantly to prevent lumps. Increase the heat to high and bring the mixture to a gentle boil.
2. Reduce the heat to low and simmer for 15-20 minutes until the vegetables are tender.

Step 5: Blend and Add Dairy

1. Use an immersion blender to puree the soup to your desired consistency (smooth or chunky). Alternatively, transfer batches to a countertop blender and blend until smooth, then return to the pot.
2. Stir in the heavy cream, whole milk, and shredded cheddar cheese. Simmer gently for another 5-7 minutes until the cheese is fully melted and the soup is creamy.
3. Add smoked paprika, dried thyme, salt, and freshly ground black pepper to taste.

Step 6: Prepare the Cheddar Drizzle

1. In a small saucepan over medium heat, combine shredded sharp cheddar cheese, heavy cream, and garlic powder.
2. Stir constantly until the mixture is smooth and well-combined. Remove from heat.

Step 7: Assemble and Serve

1. Ladle the soup into bowls.
2. Top with a handful of garlic croutons.
3. Drizzle the sharp cheddar cheese mixture over the soup.
4. Garnish with additional shredded cheddar cheese or a sprinkle of smoked paprika if desired.