Cook noodles according to package directions; drain and set aside.
Preheat oven to 350°F; grease a 9x13-inch dish.
Mix shredded chicken and pesto in a medium bowl; set aside.
Separately mix ricotta, eggs, half the Parmesan, and 1 cup mozzarella.
Melt butter in a saucepan over medium heat; whisk in flour 1 minute. Gradually add milk, stirring constantly until thickened. Season béchamel with salt, pepper, and nutmeg.
Spread one-third of béchamel in dish's bottom; layer 4 noodles, half the ricotta mixture, half the pesto chicken, and 1 cup mozzarella.
Repeat once; top with remaining 4 noodles, béchamel, mozzarella, and Parmesan.
Cover with foil; bake 35 minutes. Uncover; bake 15 minutes more, until golden. Rest 10 minutes before serving.