Melt the Butter: In a large pot over medium heat, melt the butter.
Cook Onions and Garlic: Add onions and cook until softened (about 8-10 minutes). Add garlic and cook for an additional minute.
Add Flour: Sprinkle the flour over the mixture and stir together until lightly browned (about 5 minutes).
Pour in Broth and Milk: Slowly pour in the chicken broth and milk, stirring constantly, until no flour lumps remain.
Add Vegetables and Seasonings: Add the broccoli, carrots, salt, pepper, and nutmeg. Bring the mixture to a simmer, and cook uncovered, stirring occasionally, until the broccoli and carrots are tender (24-26 minutes).
Incorporate Cheese: Reduce heat to low, and slowly stir in the cheese one cup at a time, fully melting the cheese between each addition.
Serve: Remove from heat and serve. Enjoy your delicious broccoli cheese soup!
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