Heat olive oil in a large skillet over medium heat. Add the onion, green bell pepper, and celery and cook for 5–7 minutes until softened and lightly caramelized. Add the grated garlic and cook for another 30 seconds until fragrant.
Add the ground chicken and cook until browned and fully cooked through, breaking it apart with a wooden spoon as it cooks.
Stir in the tomato paste and cook for 1 minute to deepen the flavor. Add the tomato sauce, Worcestershire sauce, yellow mustard, brown sugar, Cajun seasoning, salt, and hot sauce if using. Stir until fully combined. Reduce heat to medium-low and simmer uncovered for 8–10 minutes, stirring occasionally, until the mixture thickens into a rich sloppy joe consistency.
While the filling simmers, lightly butter and toast the brioche buns in a skillet or under the broiler until golden brown.
Pile the Cajun chicken mixture onto the toasted buns and top with pepper jack cheese, green onions, and pickles if desired. Serve warm.
