Heat 2 tablespoons of olive oil in a large skillet over medium heat. Add 1 pound of ground chicken and cook for 5–7 minutes, stirring occasionally, until no longer pink. Incorporate 1 finely chopped onion and sauté for another 3–4 minutes until translucent. Mix in 2 minced garlic cloves and 1 tablespoon of Cajun seasoning, cooking for 1 more minute until fragrant.

Stir in 2 tablespoons of tomato paste, allowing it to combine with the chicken for about 1 minute. Gradually add 1 cup of low-sodium chicken broth, 1 can of tomato sauce, and 1 tablespoon of Worcestershire sauce, mixing well. Let this simmer uncovered on low heat for 8–10 minutes, stirring occasionally until the sauce thickens. Taste and adjust salt as needed, adding 1 tablespoon of brown sugar if you prefer a touch of sweetness.

While the chicken mixture simmers, create the creamy cheese sauce. In a separate saucepan, melt 3 tablespoons of unsalted butter over medium heat. Whisk in 3 tablespoons of all-purpose flour, cooking for about 1 minute until it forms a pale roux. Gradually pour in 1 cup of whole milk, stirring constantly for 4–5 minutes, until thickened. Remove from heat and mix in 1 cup of freshly shredded sharp cheddar cheese and 1 teaspoon of paprika until melted and smooth.

Toast 4 brioche hamburger buns until golden brown. On the bottom half of each bun, generously spoon the Cajun chicken filling. Ladle the creamy cheese sauce over the filling, ensuring each generous portion is covered. Finally, sprinkle with freshly ground black pepper and cap with the top halves of the buns. Serve immediately.