Warm the oil and butter together in a large skillet over medium-low heat. Wait until the butter has melted and lightly foams without browning.

Stir in the onion, celery, and green bell pepper and cook for 7-8 minutes. The vegetables should become soft, glossy, and noticeably reduced in volume.

Add the ground chicken and break it apart with a spoon as it cooks for 5 minutes, then add the cajun seasoning and minced garlic. Keep pressing apart larger pieces until the chicken forms fine crumbles and loses its raw pink color.

Cook the seasoned chicken for 1 minute while stirring often. The garlic should smell fragrant but remain pale rather than turning brown.

Mix in the crushed tomatoes, brown sugar, hot sauce, and heavy cream, then lower the heat. The skillet should look creamy and evenly colored, with small lazy bubbles around the edges.

Let the filling cook on low for 15 minutes, stirring frequently across the bottom of the skillet. It is ready when the sauce coats the back of a wooden spoon and the chicken stays moist rather than dry.

Taste the filling and add the salt as needed. The finished flavor should be savory and warm, with enough creaminess to soften the Cajun heat.

Toast the burger rolls and spoon in the hot chicken filling to serve. The cut sides should be golden and firm enough to hold the sauce.