Heat a large Dutch oven or soup pot over medium heat. Add the bratwurst and cook for 6-8 minutes, breaking it into small pieces, until browned and cooked through. Transfer it to a plate, leaving about 1 tablespoon of rendered fat in the pot; add the butter if more cooking fat is needed.

Add the diced onion, carrots, and celery to the pot. Cook for 5-7 minutes until the vegetables begin to soften, then stir in the garlic and cook for 30 seconds.

Sprinkle the flour over the vegetables and stir continuously for 1-2 minutes. This cooks out the raw flour flavor and forms the thickening base for the soup.

Gradually whisk in the chicken broth and ale, scraping the browned bits from the bottom of the pot. Stir until the liquid is smooth and the flour is fully incorporated.

Add the cubed potatoes, paprika, thyme, salt, and black pepper. Bring the soup to a gentle simmer and cook uncovered for 15-20 minutes, or until the potatoes are tender when pierced with a fork.

Reduce the heat to low and stir in the milk or half-and-half along with the Dijon mustard. Keep the soup below a boil from this point forward.

Add the freshly shredded cheddar a handful at a time, stirring after each addition until it melts completely before adding more. Gentle heat keeps the dairy and cheese smooth.

Return the cooked bratwurst to the pot and warm it through for 2-3 minutes. Taste and adjust the salt, pepper, mustard, or ale flavor as desired.

Ladle the soup into bowls and garnish with chopped parsley or chives. Add optional toppings such as extra cheddar, croutons, green onions, a mustard drizzle, or cooked crumbled bratwurst.