Chop onion, bell pepper, and celery; slice the green onions.
Heat olive oil in a large skillet over medium-high heat.
Brown the ground chicken, breaking it up, 5–6 minutes. Season with salt and pepper.
Stir in onion, bell pepper, and celery. Cook until softened, 4–5 minutes.
Add garlic, Cajun seasoning, and smoked paprika. Cook 30 seconds until fragrant.
Pour in tomato sauce, ketchup, broth, Worcestershire, Dijon, vinegar, brown sugar, and hot sauce. Stir well.
Simmer until thick and saucy, 6–8 minutes. Stir in butter and adjust seasoning.
Toast buns. Pile the Cajun chicken onto buns and finish with green onions.