Place potatoes in a large pot. Cover with chicken broth and season with salt and black pepper. Bring to a boil. Reduce heat to medium-high and cook until potatoes are fork-tender, 20 to 25 minutes. Drain.
Heat olive oil in a skillet over high heat. Saute onion and garlic, stirring often, until fragrant, about 3 minutes. Add chicken. Season with salt, black pepper, 1/2 the basil, Cajun seasoning, and cayenne pepper. Cook until chicken is no longer pink in the center and the juices run clear, 15 to 20 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Drain chicken on a paper-towel lined plate.
Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking pan.
Place the boiled potatoes in the baking pan and mash slightly. Add chicken. Stir in evaporated milk, sour cream, butter, and remaining 1 1/2 teaspoons basil.
Bake in the preheated oven for 15 minutes. Remove from oven and sprinkle mozzarella cheese evenly over the top. Return pan to the oven and bake until cheese is melted, about 5 minutes more.
