Prepare the vegetables by washing and chopping the broccoli and carrots, and dicing the onion and garlic.
In a large pot, melt the butter over medium heat and add the diced onion and minced garlic.
Once the onion is translucent, add the shredded carrots and sauté for another 3-4 minutes.
Sprinkle in the flour and stir continuously for about 2-3 minutes to form a roux.
Gradually whisk in the vegetable broth while continuing to stir.
Bring the mixture to a gentle simmer, then add the broccoli and cook for about 15 minutes until tender.
Stir in the heavy cream and continue cooking on low heat.
Add shredded cheddar cheese and stir until melted and smooth.
Season with salt and black pepper to taste, and serve hot.