Preheat the oven to 350 degrees F (175 degrees C). Coat a 9x13-inch casserole dish with cooking spray.
Place chicken in a medium saucepan. Fill with enough water to just cover chicken. Set pan over high heat and bring to a boil. Reduce heat to medium and simmer until chicken is no longer pink in the center and juices run clear, 10 to 15 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Drain and let cool 10 minutes.
Cut cooled chicken into 1-inch pieces. Transfer to a large bowl. Add beans, sour cream, 1 cup Mexican cheese, chilies, cumin, and black pepper and mix well. Pour into the prepared casserole dish. Sprinkle with remaining cup of Mexican cheese.
Bake in the preheated oven until bubbly and cheese is melted, about 30 minutes. Let stand for 5 minutes before serving. Serve with salsa on the side.
