Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add 1 pound of ground chicken, breaking it apart and cooking until no longer pink, about 5-7 minutes. Stir continuously and set aside once done.

Stir in 1 finely chopped onion and sauté for 3-4 minutes until translucent. Add 2 minced garlic cloves and 2 tablespoons of Cajun seasoning, cooking for an additional minute until fragrant.

Incorporate 2 tablespoons of tomato paste, stirring for a minute. Pour in 1 cup of low-sodium chicken broth and 1 cup of tomato sauce, mixing well. Let simmer uncovered on low heat for 8-10 minutes until the sauce thickens.

In a small saucepan over medium heat, melt 2 tablespoons of unsalted butter, then whisk in 2 tablespoons of all-purpose flour for about a minute. Gradually whisk in 1 cup of whole milk, stirring until it thickens, about 4-5 minutes. Remove from heat and mix in 1 cup of shredded sharp cheddar and 1 teaspoon of paprika until melted.

Slice 4 brioche buns in half and butter the cut sides. Place them cut-side down in a hot pan for about 1-2 minutes until golden brown.

On each toasted bottom bun, pile a substantial amount of the Cajun chicken mixture and ladle the warm cheese sauce over. Cap with the top bun and serve immediately.