In a large pot, bring water to a rolling boil and generously salt it. Toss in the chopped broccoli stalks and florets, blanching them for 2 to 3 minutes until they turn vibrant green and are fork-tender yet retain a slight crispness. Drain the broccoli thoroughly, reserving about 1 cup of florets to use as a garnish.

Using the same pot, wipe it clean and set it over medium heat. Melt the butter until it begins to bubble gently. Add the finely chopped onions and grated carrot. Stir occasionally, allowing the vegetables to soften and become fragrant, about 4 to 5 minutes.

Sprinkle the all-purpose flour over the softened vegetables. Stir continuously to form a paste, ensuring the flour evenly coats the mixture. Let it cook for 1 to 2 minutes, then slowly pour in the vegetable stock while stirring vigorously to prevent lumps. Once all the stock is incorporated, allow the mixture to simmer gently.

Stir in the whole milk, followed by the blanched broccoli (excluding the reserved florets). Lower the heat to a gentle simmer and cook for about 10 minutes, allowing the flavors to meld together.

Using an immersion blender, puree the soup until it achieves a smooth, creamy texture. If you’re using a countertop blender, let the soup cool slightly before blending in batches, taking care not to overfill the container to avoid splashes.

Return the pureed soup to low heat. Gradually add the grated cheddar cheese, stirring until fully melted and velvety. Season with salt and freshly ground black pepper, tasting and adjusting as needed.

To serve, ladle the soup into bowls, garnishing with the reserved broccoli florets and a sprinkle of extra cheese. Pair with your favorite bread for dipping and enjoy this comforting dish.