Melt the butter in a large pot or Dutch oven over medium heat.
Add the onion, salt, and several grinds of pepper. Cook, stirring, for 5 minutes or until softened.
Stir in the garlic and cook for another minute, then sprinkle in the flour. Whisk continuously for 1 to 2 minutes, or until the mixture is golden and thickened.
Slowly pour in the milk, whisking as you go to prevent clumping. Add the vegetable broth, broccoli, carrot, and mustard. Stir to combine.
Reduce the heat to low, cover, and simmer for 15 to 20 minutes until the broccoli is tender.
Gradually stir in the shredded cheddar cheese, one handful at a time, until the soup becomes creamy and smooth. Be sure to keep the heat low to avoid separating the cheese.
Give it a final taste and adjust the seasoning as needed. Serve your soup topped with homemade croutons, if desired.