In a Dutch oven (or a large pot with a lid) over medium heat, melt the butter.
Add onions and cook until softened (about 5-8 minutes). Add garlic and cook for an additional minute.
Sprinkle the flour over the mixture and stir together until lightly browned (about 5 minutes).
Slowly pour in the chicken broth and milk, stirring constantly, until no flour lumps remain.
Add the broccoli, carrots, salt, pepper, nutmeg, and cheese. Bring the mixture to a simmer. Cook uncovered, stirring occasionally, until the cheese has melted. Reduce heat to medium-low.