Add a thin layer of sauce to the bottom of 2 gratin dishes. For each dish, top each with a pasta sheet, cutting if needed to fit the dish, followed by a thin layer of mozzarella. Crumble 1/4 of the ground chicken over the mozzarella. Add a few small dollops of goat cheese on top, and evenly distribute some spinach on top. Top with an even layer of sauce.
Repeat until all of the ingredients, but save some mozzarella for the last layer. Bake until bubbly and browned on top, about 30 minutes. Let rest for 5-10 minutes before eating.