In a large nonstick skillet over medium-low heat, heat 1 tablespoon oil. Add onion; cook, stirring occasionally, until softened, about 5 minutes. Stir in garlic and cook, stirring, 1 minute. Stir in mushrooms, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Increase heat to medium and cook, stirring occasionally, until mushroom liquid evaporates, 10 to 15 minutes. Remove mushroom mixture to a bowl.
In a skillet over medium-high heat, heat remaining 1 tablespoon oil. Stir in radicchio and 1/4 teaspoon salt; cook, stirring frequently, until radicchio wilts and liquid evaporates, about 5 minutes. Stir in sage and cook 1 minute. Return mushroom mixture to skillet and stir to combine; set aside.
Place flour in a large saucepan. Gradually whisk in milk until smooth. Over medium heat, cook, stirring frequently, until sauce comes to a boil. Reduce heat to low and simmer 3 minutes, stirring constantly. Remove pan from heat; stir in 1/4 cup Parmesan cheese, nutmeg, 1/2 teaspoon salt, and 1/8 teaspoon pepper. Set aside.
Preheat oven to 375°. Spray a 13" x 9" baking dish with cooking spray. In a small bowl, combine ricotta and mozzarella cheeses; stir well.
Reserve 1 cup sauce for top layer. Spread 1/2 cup sauce in bottom of prepared dish. Arrange 4 lasagna noodles over sauce, overlapping noodles slightly to fit. Spread with 1/2 cup ricotta mixture. Spoon on half of mushroom mixture. Top with half of chicken. Pour half of remaining sauce over chicken. Repeat layering. Top with remaining lasagna noodles and spread with reserved 1 cup sauce.
Coat a sheet of foil with cooking spray and cover baking dish. Bake lasagna 35 minutes. Uncover dish and sprinkle with remaining 1/4 cup Parmesan cheese. Bake uncovered 15 minutes. Let stand 15 minutes.