Melt the butter in a soup pot, then add the diced onion. Saute on medium heat until translucent (about 10 minutes). Season with salt, pepper, and a pinch of cayenne. Add the minced garlic and continue to saute until fragrant. Add the grated carrots, and after five minutes, stir in the flour.

Cook for one minute, then add the chicken broth and the broccoli. Mix well. After five minutes, add the half-and-half. Mix well and let it come up to a simmer. Turn off the heat and add the cheese. Stir gently until the cheese is fully incorporated. Let it melt for one minute. Serve and enjoy!