Preheat the oven to 375°F (190°C). Grease a 9×13-inch baking dish or spray with cooking spray.
If any chicken breasts are thicker than 1/2 inch, place them between two sheets of plastic and pound to about 1/2-inch thickness.
In a medium bowl, combine the softened cream cheese, drained chopped spinach, 3 tablespoons grated Parmesan, and 1 cup shredded mozzarella until smooth and evenly mixed.
Spread an even layer of the cheese and spinach mixture onto each chicken breast, then roll each breast up tightly and secure with a toothpick.
Place the crushed RITZ crackers mixed with the remaining grated Parmesan in a shallow dish. Put the beaten egg in a separate shallow bowl.
Dip each rolled chicken first into the beaten egg, then roll in the cracker-and-Parmesan mixture to coat evenly.
Place the coated chicken rolls seam-side down in the prepared baking dish and spray the tops lightly with cooking spray.
Bake in the preheated oven for 30 minutes, until the chicken is cooked through and the coating is golden.
Remove from the oven, take out the toothpicks, spoon warmed pasta sauce over each roll, and sprinkle with the remaining shredded mozzarella. Serve immediately.