Preheat oven to 350℉. Melt the butter over medium heat in a large pot or Dutch oven. Add the onion and garlic, and season with salt and pepper. Cook for 5 minutes, stirring occasionally, until softened.
Sprinkle in flour and whisk for 1 - 2 minutes, or until golden. Next, slowly add in your milk, whisking as you add it.
Now add in your broth, Dijon, carrot, and broccoli. Season with salt and pepper. Simmer for 15 - 20 minutes or until the broccoli is tender. While this simmers, make your croutons.
Tear your Italian loaf into rough bite-sized chunks on a baking sheet lined with parchment paper. Drizzle with olive oil and season liberally with salt and pepper. Toss and then evenly spread out the pieces so they don't touch.
Bake croutons for 10 - 15 minutes, depending on how crispy you like them.
Finally, turn the heat off on the soup and sprinkle in your cheese, stirring to combine. Taste and adjust seasoning if needed.