1Melt the butter in a large pot or Dutch oven over medium heat.
2Add the onion, salt, and several grinds of pepper and cook, stirring, for 5 minutes, or until softened.
3Stir 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.
4Slowly pour in the milk, whisking continuously.
5Add the broth, broccoli, carrot, and mustard and stir to combine.
6Simmer for 15 to 20 minutes, or until the broccoli is tender.
7Gradually add the cheese, stirring after each addition, until all the cheese is melted and the soup is creamy.
8Season to taste and serve with croutons, if desired.