In a Dutch oven, melt the butter over medium heat. Add the chopped onions and sauté until they become soft and translucent. Stir in the minced garlic and continue cooking for an additional 2 minutes.

Gradually incorporate the flour, cooking it for 2 to 3 minutes while stirring constantly. This step ensures the flour loses its raw taste and develops a smooth base.

Slowly pour in the chicken broth and whole milk, stirring continuously to prevent any lumps from forming. Once the mixture is smooth, add the broccoli florets and grated carrots. Stir well, then cover and allow the soup to simmer on low heat for about 15 minutes or until the vegetables are tender.

Add the heavy cream and shredded cheddar cheese to the pot, stirring until the cheese melts completely. Taste the soup and adjust the seasoning with salt and pepper as necessary. Let the soup cook for an additional 2 minutes to fully incorporate the flavors.