Melt the butter in a large pot or Dutch oven over medium heat. Add the onion along with a pinch of salt and a few grinds of black pepper. Cook, stirring occasionally, for about 5 minutes, or until the onion softens. Stir in the garlic and cook for 1 minute more. Sprinkle in the flour and whisk constantly for 1 to 2 minutes, until it turns lightly golden. Slowly pour in the milk, whisking all the while to avoid lumps.

Next, add the broth, broccoli, carrot, and mustard. Stir well and bring the mixture to a gentle simmer. Cook for 15 to 20 minutes, or until the broccoli is tender.

Gradually stir in the cheese, a handful at a time, allowing it to melt fully before adding more. Continue until the soup is smooth, creamy, and rich. Taste and adjust the seasoning as needed. Serve hot, topped with crunchy croutons if you like.