Melt butter in a medium saucepan over medium heat. Add garlic and cook 1 minute. Whisk in flour and cook 1 minute. Slowly whisk in milk, then cook, whisking often, until thickened, 4 to 5 minutes. Off heat, stir in nutmeg, salt, and pepper. Set aside.

Stir together ricotta, egg, about 3 tablespoons of the parmesan, and the squeezed spinach. Season with salt and pepper.

Preheat oven to 375°F (190°C). Grease a 9x13-inch baking dish. Spread a little béchamel on the bottom. Layer 3 noodles, half the ricotta mixture, ⅓ of the chicken, ⅓ of the mozzarella, and ⅓ of the remaining béchamel. Repeat: 3 noodles, remaining ricotta, another ⅓ chicken and mozzarella, another ⅓ béchamel. Top with last 3 noodles, remaining chicken, last of the béchamel, then remaining mozzarella and parmesan.

Cover with foil and bake 25 minutes. Uncover and bake 10 to 15 minutes more, until golden and bubbly. Broil 1 to 2 minutes if you want a darker top.

Let the lasagna rest 10 minutes, then slice and serve.