In a large Dutch oven, fry bacon over medium heat until crisp, about 10 minutes. Using a slotted spoon, remove the bacon to a paper towel-lined plate. Reserve 1 tablespoon of bacon drippings in the pot.
Add the butter to the pot with the bacon drippings. Increase the heat to medium-high. When the butter melts, add the onion and cook, stirring regularly, until starting to soften, about 3-5 minutes. Sprinkle flour over the onion and cook, stirring constantly, for 1-2 minutes. Slowly whisk in the chicken broth until incorporated.
Add broccoli and carrots. Bring to a boil, then reduce the heat and simmer (uncovered) until the soup thickens slightly and the vegetables are tender, about 15-20 minutes.
Reduce the heat to the lowest temperature. Stir in the cheddar, cream cheese, beer, Dijon, hot sauce, paprika, and bacon (reserve some of the cooked bacon if you'd like to use it for a garnish on top of your soup). Whisk gently until all of the cheese melts, about 5-7 minutes. Do not let the mixture boil. Taste and season with salt and pepper, if desired. Ladle into individual bowls and garnish with optional toppings.