Heat olive oil in a large skillet over medium-high heat. Add the diced onion, bell pepper, and celery. Sauté for 5-7 minutes until softened.

Add the minced garlic and Cajun seasoning to the skillet. Stir and cook for 1 minute until fragrant.

Add the ground chicken, breaking it apart with a wooden spoon. Cook for 6-8 minutes until browned and cooked through.

Stir in the tomato paste, coating the meat and vegetables. Cook for 1 minute to caramelize slightly.

Pour in the chicken broth, scraping up any browned bits from the bottom of the pan. Reduce heat and simmer for 3-5 minutes until the liquid reduces and thickens.

Remove the skillet from heat. Stir in the heavy cream and shredded cheddar cheese until completely melted and smooth.

Toast the brioche buns. Spoon a generous amount of the creamy Cajun chicken mixture onto the bottom half of each bun, cover with the top half, and serve immediately.