Melt the butter in a large Dutch oven or heavy pot over medium heat.
Add the onion and carrots. Cook for about 5 minutes, stirring occasionally, until softened.
Add the garlic and cook for 30 seconds.
Sprinkle the flour over the vegetables and stir constantly for 1 minute.
Slowly pour in the broth while stirring so the flour mixture stays smooth.
Add the chopped broccoli and bring the soup to a gentle simmer.
Cook for 10 to 12 minutes, or until the broccoli is tender.
Reduce the heat to low. Stir in the milk, heavy cream, Dijon mustard, paprika, salt, and black pepper.
Warm gently for 3 to 5 minutes. Do not boil.
Turn the heat to its lowest setting or remove the pot from the heat.
Add the shredded cheddar one handful at a time, stirring until completely melted before adding the next handful.
Taste and adjust the salt and pepper.
If the soup is thicker than you like, stir in a splash of milk or broth.
Serve warm.