The Best Broccoli Cheddar Soup

Gather all of your ingredients.
Dice the onion and 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 sauté for 2 minutes more, or until the flour is coating the bottom of the pot and is a light golden brown color.
Pour in chicken broth and whisk to dissolve the flour. Turn the heat up to medium-high and, while stirring often, bring the broth to a simmer. When it reaches a simmer it will thicken to a gravy consistency.
Turn the heat down to medium and whisk in the whole milk, smoked paprika, garlic powder, black pepper, and cayenne pepper. Allow the broth to come back to a simmer.
Add the chopped broccoli and carrots and continue to simmer the vegetables, stirring often, for 15 minutes or until the carrots are tender.
Test a piece of carrot or broccoli to make sure it's to your desired tenderness. If not, simmer a bit longer.
Turn the heat 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).