Melt 1-2 tablespoons of butter in a large pot over medium-high heat. Once the butter is bubbly, add diced onion and sauté for about 3 minutes until it becomes translucent and fragrant.

Stir in diced carrots along with a pinch of salt and pepper. Cook for an additional 3-4 minutes, stirring occasionally, until the carrots soften and begin to brighten.

Incorporate minced garlic into the pot, stirring for approximately 30 seconds until you can smell its fragrant aroma.

Add peeled and chopped potatoes to the pot, followed by chicken stock to cover the ingredients. Bring the mixture to a gentle boil, then reduce the heat, cover, and let it simmer for about 10 minutes.

Carefully stir in the broccoli florets, letting them mingle with the other ingredients. Cover the pot again and let it simmer for another 10 minutes, or until both the broccoli and potatoes are fork-tender.

In a small bowl, mix cornstarch with milk until it forms a smooth slurry. Stir this mixture gently into the hot soup, continuing to simmer for a few minutes until the soup thickens to a velvety consistency.

Once thickened, stir in the shredded sharp cheddar cheese bit by bit. Allow the cheese to melt completely, creating a rich, creamy finish to your soup.

Ladle the soup into bowls and serve hot, garnished with extra cheese or a sprinkle of black pepper if desired.