Start by melting the butter in a large pot over medium heat. Add the diced onion and carrots and cook until the onion becomes soft and translucent. This step builds the base flavor and brings out natural sweetness from the vegetables.
Stir in the minced garlic and cook briefly until fragrant. Sprinkle the flour over the vegetables and stir continuously for about a minute. This creates a roux that will thicken the soup evenly.
Slowly pour in the vegetable broth while stirring to avoid lumps. Add the half and half, broccoli, Dijon mustard, salt, and black pepper. Stir well to combine all the ingredients smoothly.
Bring the soup to a gentle simmer and cook until the broccoli becomes tender but still vibrant in color. Stir occasionally to ensure even cooking and prevent sticking.
Reduce the heat to low and gradually add the grated cheddar cheese. Stir constantly until the cheese melts completely and the soup becomes creamy and smooth. Avoid boiling to keep the texture silky.