Bring a large pot of salted water to a boil, and blanch the broccoli florets for 2-3 minutes until soft and bright green in color. Drain and set aside.

Melt the butter in a large stock pot over low-medium heat. Add the onions and carrots. Cook for 3-5 minutes until they begin to soften. Add the garlic, and cook for an additional 30 seconds.

Sprinkle the flour into the pan, and cook for 2-3 minutes to remove any raw flour taste. Slowly whisk in the chicken broth, heavy cream, and milk. Continue to whisk until the mixture  thickens slightly and begins to steep. Don’t let the liquid boil.

While whisking, add the shredded cheese little bits at a time. Make sure to whisk well between each addition to ensure all the cheese is melted.

Taste the soup, and add salt and pepper if needed.

Serve hot with extra broccoli and cheese on top. See post for storage options.