Melt 3 tablespoons of unsalted butter in a large pot over medium heat. Add 1 chopped yellow onion and a pinch of sea salt and black pepper. Sauté for about 5 minutes until the onion is soft and translucent.

Stir in 2 chopped garlic cloves and cook for an additional minute until aromatic.

Sprinkle in ¼ cup of all-purpose flour, whisking constantly for 1-2 minutes until lightly golden to create a roux.

Gradually pour in 4 cups of whole milk while whisking to prevent lumps, then add 4 cups of vegetable broth and stir until smooth.

Introduce 4 cups of broccoli florets and 1 julienned carrot, stirring well. Bring to a gentle simmer and cook for 15-20 minutes until tender.

Add 1 tablespoon of Dijon mustard, stirring to combine, and adjust seasoning with salt and pepper.

Reduce heat to low and slowly add 2 cups of shredded sharp cheddar cheese, stirring until melted and creamy.

Ladle the soup into warm bowls and serve hot, optionally topping with croutons or crusty bread.