In a large pot or Dutch oven, melt the butter over medium heat.
Add the chopped onion, a pinch of sea salt, and freshly ground black pepper. Cook until the onion is soft and fragrant, about 5 minutes.
Stir in the chopped garlic and cook for an additional minute.
Sprinkle the flour over the onion and garlic mixture. Whisk continuously for 1 to 2 minutes until the flour takes on a light golden color.
Slowly pour in the milk while whisking constantly to prevent lumps.
Once the mixture is smooth, gradually stir in the vegetable broth.
Add the chopped broccoli, julienned carrot, and Dijon mustard to the pot. Stir and let it simmer for 15 to 20 minutes until fork-tender.
Turn the heat to low, gradually add the shredded cheddar cheese, stirring between additions until melted.
Adjust seasoning with salt and pepper, serve warm, and top with homemade croutons if desired.