Melt butter in a skillet over medium heat. Cook and stir chicken until no longer pink in the center and the juices run clear, about 3 to 4 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Set aside to cool until cool enough to handle; cut into chunks.
Mix chicken, cream cheese, mushrooms, green onion, salt, and black pepper together in a bowl.
Unroll crescent roll dough. Place a portion of the chicken mixture onto the center of each piece of dough and fold the three corners up, forming a pocket. Pinch edges to seal.
Bake in preheated oven until golden brown, about 20 minutes.