Melt the butter in a large Dutch oven or pot over medium heat.
Add the chopped onion and cook for 3 minutes or until translucent. Then add the minced garlic and cook for 1 minute.
Add the flour and whisk to form a paste.
While still whisking, pour in the milk, half-and-half, or heavy cream. Then add the chicken stock and bring it to a boil.
Once it is boiling, add the chopped broccoli and carrots.
Cover and simmer on low heat for 20-25 minutes.
After simmering, uncover and season with salt and pepper to taste.
Turn off the heat, add 2 cups of the shredded sharp cheddar cheese and stir until it melts.
Serve with more cheese on top, crackers or toasted bread.