Begin by dicing one onion and mincing two cloves of garlic. Next, peel and dice two medium-sized carrots and two medium potatoes, then chop about a cup of broccoli into florets.

In a large pot, melt three tablespoons of butter over medium-low heat. Add the diced onion and minced garlic, and cook for about 7 to 10 minutes until the onions are translucent and fragrant.

After the onions and garlic have softened, add the diced carrots to the pot. Sauté for an additional 5 minutes, stirring occasionally.

Stir in the diced potatoes and pour in approximately 4 cups of vegetable broth. Bring to a boil, then reduce to a gentle simmer. Cook for about 15 minutes, until the potatoes are tender.

In a separate bowl, whisk together one cup of milk and three tablespoons of flour until smooth. Gradually stir this into the boiling soup, allowing it to thicken.

Add the chopped broccoli, stirring to incorporate it. Let it simmer for an additional 10 to 15 minutes, until the broccoli is tender.

Remove from heat and stir in one cup of grated sharp cheddar cheese until it melts completely. Taste and season with salt and pepper.

Ladle the soup into bowls and garnish with freshly chopped parsley or chives. Drizzle with olive oil if desired.