Melt Butter. Start by melting butter in a Dutch oven or large pot over medium heat.
Saute Onion. Add the onion to the melted butter and stir, 5-7 minutes, or until onion is softened and turning translucent.
Add Flour & Seasonings. Next, sprinkle in the gluten-free flour, salt, pepper, paprika, and garlic. Stir constantly for 3-4 minutes over medium heat, or until the flour has browned slightly and the mixture looks golden. (This helps remove any floury taste, so don't rush this step!)
Mix In Broth and Half & Half. Pour in the broth and half and half and stir to combine. Heat just until the mixture starts to bubble, then reduce heat to a simmer to thicken slightly, about 5-7 minutes.
Add Vegetables. Then, add the broccoli and carrots. Cook just 6-8 minutes or so, until the broccoli is softened and the soup is quite thick.
Mix In Cheese a little at a time, stirring to fully melt the cheese before adding the next handful. (I add mine in 3-4 increments)
Finish The Soup by stirring in hot sauce. Garnish with a little more pepper and cheese, if desired.
Serve & Store. Enjoy the broccoli cheddar soup right away, or store leftovers in an airtight container in the fridge 3-4 days. (This thick soup will get even thicker in the fridge, but will loosen back up when you reheat it.)