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.
Notes
*Use either heavy cream, half-and-half, or milk depending on how thick you want the soup's texture. Keep in mind that it will thicken as it cools.
Chop the broccoli into bite-sized pieces to make it more comfortable to eat.
You can shred or chop the carrots yourself. But buying a bag of matchstick carrots from the grocery store is inexpensive and convenient.
Grate the cheese yourself to get the creamiest texture. Pre-shredded cheese has anti-caking agents that prevent the cheese from melting as smoothly. This may also result in a grainy texture.
Use sharp cheddar, mild/medium cheddar, white cheddar, or a combination of all three.