Preheat oven to 400 degrees F (200 degrees C). Grease 2 baking sheets with cooking spray.
Melt butter in a saucepan over medium heat. Add chopped green onions and cook until tender, about 4 minutes. Add garlic and cook for 1 minute more. Mix in cooked chicken, salt, nutmeg, and black pepper; cook 1 minute more.
Arrange pastry discs on a flat work surface. Dab a little water around the rims. Place a large tablespoonful of the chicken mixture in the center; fold pastry over and press with a fork to seal.
Transfer empanadas to the prepared baking sheets. Brush with beaten egg.
Bake in the preheated oven until golden, about 20 minutes.