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Dice the onion. Chop the broccoli into very small pieces. Peel and slice the carrots.
Add the butter and onion to a large soup pot and sauté over medium until the onions are soft and translucent. Add the flour and continue to sauté for about 2 minutes more, or until the flour is coating the bottom of the pot and is a light golden brown color.
Add the chicken broth and whisk to dissolve all the flour off the bottom of the pot. Turn the heat up to medium-high and, while stirring often, allow the broth to come up to a simmer. When it reaches a simmer it will thicken to a gravy consistency.
Turn the heat back down to medium and whisk in the milk, smoked paprika, garlic powder, black pepper, and cayenne pepper. Allow the broth to come back up to a simmer.
Add the chopped broccoli and carrots, allow the broth to come back up to a simmer, and continue to simmer the vegetables, stirring often, for 15 minutes, or until the carrots are tender.
Finally, turn the heat down to medium-low and stir the shredded cheese into the soup, one handful at a time, until it is fully melted. Taste the soup and add salt if needed (this will depend on the salt content of your broth—I did not add any).