Melt the butter in a large pot over medium heat.

Add the onion, salt, and several grinds of pepper and cook, stirring, for 5 minutes, or until softened.

Stir in the garlic and cook for another minute, then sprinkle in the flour and whisk continuously for 1 to 2 minutes, or until the flour turns golden.

Slowly pour in the milk, whisking continuously.

Add the broth, broccoli, carrot, and stir to combine. Simmer for 15 to 20 minutes, or until the broccoli is tender.

Gradually add the cheese, stirring after each addition, until all the cheese is melted and the soup is creamy.