Cook lasagna noodles according to package directions, drain, and set aside.
Preheat oven to 350°F and grease a 9x13-inch baking dish.
In a medium bowl, combine shredded chicken with pesto and set aside.
In another bowl, mix ricotta cheese, eggs, half the Parmesan, and 1 cup mozzarella.
Make béchamel sauce by melting butter in a saucepan over medium heat, whisking in flour for 1 minute, then gradually adding milk while stirring constantly until thickened. Season with salt, pepper, and nutmeg.
Spread one-third of the béchamel sauce in the bottom of the baking dish, then layer 4 noodles, half the ricotta mixture, half the pesto chicken, and 1 cup mozzarella.
Repeat layers once more, then top with remaining 4 noodles, remaining béchamel sauce, remaining mozzarella, and remaining Parmesan.
Cover with foil and bake for 35 minutes, then uncover and bake 15 minutes more until golden. Let rest 10 minutes before serving.