In a large pot over medium-low heat, melt the butter until it’s bubbling gently, then add the diced onion and minced garlic. Sauté for about 7–10 minutes until the onion becomes translucent and fragrant.

Next, stir in the diced carrots and continue to sauté for an additional 5 minutes until the carrots soften slightly.

Add the diced potatoes and pour in the vegetable broth, bringing the mixture to a rolling boil. Reduce the heat to a simmer and cover loosely, cooking for 10–12 minutes.

Whisk together the milk and flour in a small bowl until smooth. Gradually stir this mixture into the pot, stirring continuously to avoid lumps.

Stir in the finely diced broccoli and allow it to simmer for another 10–15 minutes on low heat until tender.

Add the freshly grated sharp cheddar cheese, stirring continuously until it melts completely.

Ladle the soup into warm bowls, garnishing with extra cheese or a sprinkle of cracked black pepper if desired. Serve hot.