In a large soup pot or Dutch oven, melt the butter with the olive oil over medium heat. Saute the shredded onions and carrots in the fat for 5-6 minutes, or until the onions are translucent. At this point add garlic and let cook for one minute.

Add in flour, and stir with a wooden spoon or silicone spatula to cook out the raw flour, about a minute or so. Make sure to stir constantly so the flour doesn't stick to the bottom.

Gradually add the chicken stock to your roux, about ¼ cup at a time, mixing constantly. It will seize up before it gets smooth, so don't worry if it looks clumpy as you stir in the stock. Finish adding in the stock, then add the broccoli florets to the pot.

Throw in the kosher salt, black pepper, cayenne pepper, mustard powder (if using), and paprika. Then add half and half or the milk of your choice. Stir well, then turn the heat up to medium-high until soup begins to come to a lively simmer.

Let cook for 13-15 minutes, or until the broccoli florets are completely tender. If you want a smooth soup, puree either some of the soup or the entire pot of soup with an immersion blender before adding the cheese.

Turn the heat off, and add the cheddar cheese to your soup. Stir it in until completely combined, handful by handful, then taste it (super important! Don't want bland soup, right?). Season with salt and pepper to taste, then serve hot in soup bowls.