Sauté Aromatics: In a large heavy-bottomed pot or Dutch oven, melt the butter over medium heat. Add the chopped onion, sea salt, and a few grinds of black pepper. Cook, stirring frequently, until the onions are soft and translucent, about 5 minutes. This step builds a flavorful base for the soup.
Infuse with Garlic and Thicken: Add the minced garlic to the softened onions and cook for an additional minute until fragrant. Sprinkle the flour evenly over the mixture, stirring constantly to combine. Continue cooking for 1–2 minutes, allowing the flour to toast slightly, which will help thicken the soup and develop a nutty flavor.
Gradually Add Liquids: Slowly pour in the milk while whisking continuously to prevent lumps. Once combined, stir in the vegetable broth. The liquid mixture should be smooth and start to thicken as it heats.
Simmer Vegetables: Add the broccoli florets, sliced carrot, and Dijon mustard to the pot. Stir everything together, then reduce the heat to low and let the soup simmer gently for 15–20 minutes. The vegetables should become tender but still maintain some texture.
Melt Cheese and Finish: Remove the pot from heat and gradually stir in the shredded cheddar cheese, allowing it to melt fully before adding more. This gradual melting prevents the cheese from clumping and ensures a creamy finish. Taste and adjust seasoning with salt and pepper as needed.
Serve Warm: Ladle the soup into bowls and serve immediately, optionally topped with homemade croutons or paired with warm crusty bread for added crunch and heartiness.