Melt butter in a Dutch oven over medium heat. Add onion, carrot, salt and pepper; cook about 5 minutes until softened. Add garlic and cook about 1 minute until fragrant. Add flour and cook 1 to 2 minutes, stirring constantly, until golden brown and forms a paste. Add broth is a slow stream, stirring constantly with a whisk. Add milk, broccoli and mustard; bring to a simmer over medium-high heat. Reduce heat to medium-low and continue cooking about 15 minutes. stirring occasionally, until the broccoli is tender. Reduce heat to low. Stir in cheese, a handful at a time, until melted. Ladle into bowls and garnish with additional cheese and pepper, if desired.