Melt the butter in a large Dutch oven over medium heat. Add the onion and cook until softened, 4 to 5 minutes. Add the garlic and cook 30 seconds.
Sprinkle in the flour and whisk constantly for 1 to 2 minutes.
Slowly whisk in the chicken broth, then the milk and half-and-half. Bring to a gentle simmer and cook, whisking often, until slightly thickened, about 5 minutes.
Stir in the broccoli, carrot, salt, pepper, paprika, and nutmeg. Simmer uncovered until the broccoli is tender, 10 to 12 minutes.
Reduce the heat to low. Add the cheddar one handful at a time, stirring until melted before adding more. Stir in the Dijon mustard.
Taste, adjust the seasoning, and serve hot topped with extra cheddar.