Preheat oven to 375F. Lightly grease a 9x13 inch baking dish with olive oil or cooking spray.

Heat olive oil in a large skillet over medium heat. Add garlic and cook 30 seconds until fragrant. Add the squeezed-dry spinach and cook 2 minutes, stirring, to evaporate any remaining moisture. Remove from heat and stir in the shredded chicken. Season with 1/2 teaspoon salt and 1/4 teaspoon black pepper.

In a medium saucepan, melt butter over medium heat. Whisk in flour and cook 1 minute, stirring constantly, until the mixture turns pale golden and smells nutty. Do not let it brown.

Slowly whisk in warm milk and chicken broth, pouring in a steady stream while whisking to prevent lumps. Bring to a gentle simmer and cook 6-8 minutes, whisking occasionally, until the sauce thickens enough to coat the back of a spoon. Remove from heat and stir in 3/4 cup Parmesan, nutmeg, 1/2 teaspoon salt, and 1/4 teaspoon black pepper.

Spread 3/4 cup sauce on the bottom of the prepared dish. Arrange 4 noodle sheets over the sauce, breaking to fit as needed. Spread half the ricotta over noodles, then half the chicken-spinach mixture, 1 cup sauce, and 1/2 cup mozzarella. Repeat layers once more, then top with final 4 noodles, remaining sauce, remaining mozzarella, and reserved 1/4 cup Parmesan.

Spray a sheet of foil with cooking spray and cover the dish tightly, sprayed side down. Bake 30 minutes. Uncover and bake 15-20 minutes more until the top is bubbly and spotted golden brown. Let rest 10 minutes before slicing. Garnish with fresh basil.