Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add 1 pound of ground chicken, breaking it apart with a spatula. Cook for 5-7 minutes until the chicken is fully cooked and no longer pink, stirring occasionally.

Stir in 1 finely chopped onion, cooking for 3-4 minutes until it becomes translucent. Add 2 minced garlic cloves and 2 tablespoons of Cajun seasoning. Cook for another minute, stirring constantly.

Add 2 tablespoons of tomato paste to the skillet and mix well. Pour in 1 cup of low-sodium chicken broth, 1 cup of canned tomato sauce, and 1 tablespoon of Worcestershire sauce. Let the mixture simmer uncovered for 8-10 minutes until it thickens slightly.

In a medium saucepan, melt 2 tablespoons of unsalted butter over medium heat. Whisk in 2 tablespoons of all-purpose flour, stirring constantly. Gradually add 1 cup of whole milk, whisking until the mixture thickens. Lower the heat and stir in 1 cup of freshly shredded sharp cheddar cheese and 1/2 teaspoon of paprika.

Butter the cut sides of 4 brioche hamburger buns. Toast them butter-side down in a hot skillet or under the broiler for 1-2 minutes until golden brown.

Spoon the Cajun chicken mixture onto the bottom half of each toasted bun. Drizzle the creamy cheese sauce over the chicken, sprinkle with freshly ground black pepper, and cap with the top bun.