Cook the chicken: Heat olive oil in a large skillet over medium-high heat. Add ground chicken and cook for 5-7 minutes, breaking it apart with a wooden spoon, until no pink remains and chicken is lightly browned.
Add aromatics: Stir in diced onion and cook for 3-4 minutes until softened and translucent. Add minced garlic and Cajun seasoning, cooking for 1 minute until fragrant.
Build the sauce: Mix in tomato paste and cook for 1 minute until the meat is coated in a brick-red glaze. Pour in chicken broth, tomato sauce, and Worcestershire sauce. Stir to combine.
Simmer: Reduce heat to low and simmer uncovered for 8-10 minutes until the mixture thickens and becomes spoonable. Taste and season with salt. Add brown sugar if desired to balance heat.
Toast buns: Butter the cut sides of brioche buns and toast in a hot skillet or under the broiler for 1-2 minutes until golden brown.
Assemble and serve: Spoon generous amounts of Cajun chicken filling onto bottom buns. Grind fresh black pepper over top, cap with toasted top buns, and serve immediately.