Heat butter (or olive oil) in a large stockpot over medium-high heat until melted. Add onion and carrots and saute for 4-5 minutes, stirring occasionally, until both are cooked through and the onion is translucent. Stir in the garlic and saute for an additional 1 minute, then stir in the flour and saute for an additional 1 minute. Whisk in the chicken stock until it is evenly combined. Stir in the milk, broccoli, mustard, salt and black pepper until combined. Continue cooking, stirring every few minutes, until the soup reaches a simmer. Reduce heat to medium and continue to simmer the soup for another 6-8 minutes, until the broccoli is cooked through and soft. Stir in the freshly-grated cheddar cheese until it is completely melted and evenly mixed into the soup. Serve immediately, topped with extra cheese if desired. *Be sure to freshly-grate the cheddar (yourself) from a block of cheese for this recipe. The store-bought shredded bagged cheeses have a coating on them that makes it difficult for the cheese to smoothly melt, so it's better to grate the cheese yourself. If you don't have a grater, you can also just dice the cheese into small cubes which will also melt well.