In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion, along with salt and a dash of black pepper. Sauté for about 5 minutes until the onion becomes tender.
Next, mix in the chopped garlic and cook for an additional minute. Sprinkle in the flour and whisk continuously for 1 to 2 minutes, allowing it to reach a rich golden hue.
Gradually pour in the milk while whisking to blend the mixture smoothly.
Add the vegetable broth along with the broccoli, carrot, and Dijon mustard. Stir to blend all ingredients together, then allow it to simmer for 15 to 20 minutes, or until the broccoli is cooked to tenderness.
Slowly incorporate the shredded cheddar cheese, mixing thoroughly after each addition until the cheese is completely melted and the soup is creamy. Adjust seasoning to taste.
Serve the soup hot, garnished with homemade croutons for an extra crunchy texture!