In a large skillet, melt 1-2 tablespoons of butter over medium-high heat. Once bubbly, add the diced onion and sauté for about 3 minutes until soft and translucent.

Stir in the diced carrots along with a pinch of salt and ground pepper. Continue cooking for another 3-4 minutes until the carrots soften.

Add minced garlic to the skillet and sauté for an additional 30 seconds, avoiding browning the garlic.

Incorporate peeled and chopped potatoes into the skillet, followed by the chicken stock. Bring to a robust simmer, cover, and cook for about 10 minutes.

After the initial simmer, add broccoli florets. Simmer uncovered for an additional 10 minutes or until tender.

In a small bowl, whisk together cornstarch and milk until smooth. Slowly pour this into the simmering soup, stirring continuously to avoid lumps.

Stir in the shredded sharp cheddar cheese until melted, then taste and adjust seasoning.

Ladle the soup into bowls and serve hot, preferably with crusty bread or a fresh salad.