To make the béchamel sauce melt the butter in a medium saucepan over medium-low heat. Add the flour and whisk for 3 minutes. Whisk in the milk. Increase the heat to medium-high. Whisk the sauce until it comes to a simmer and is thick and smooth, about 3 minutes. Whisk in the salt, pepper, and nutmeg and remove from the heat.
Preheat the oven to 450ºF.
In a medium bowl, combine ricotta, spinach, 1 cup Parmesan, ½ cup Fontina, chicken, egg, salt, and pepper.
Bring a large pot of salted water to a boil. Add the noodles and cook just until tender but still firm to the bite. Drain and run briefly under cool water. Arrange the noodles in a single layer on a towel or baking sheet to prevent them from sticking together.
Grease a 9x13 glass baking dish. Pour the béchamel sauce over the bottom of the dish. Spread large spoonfuls of the ricotta mixture evenly over each noodle. Starting at one end, roll each noodle up like a jelly roll. Lay the lasagna rolls seam side down, without touching, on top of the sauce in the dish. Spoon 1 cup of marinara sauce over the lasagna rolls. Sprinkle with the remaining Fontina, Parmesan, and mozzarella. Cover tightly with foil and bake until heated through, about 20 minutes. Uncover and bake until the cheese on top is golden, about 15 minutes longer. Let stand for 10 minutes. Meanwhile, heat the remaining marinara sauce in a small saucepan over medium heat until hot, and serve alongside.