Preheat oven to 450 degrees F. Spray a 9 x 13-inch baking dish with cooking spray and set aside.
In a large skillet, heat olive oil over medium-high heat. When oil shimmers, add mushrooms and onion; cook until tender (about 6-7 minutes). Add garlic; cook 1 minute longer. Add flour to the skillet and stir until blended. Stir in chicken and Alfredo sauce. Bring to a boil. Reduce heat; simmer uncovered for about 5 minutes (to allow the sauce to thicken slightly).
In a separate bowl, stir together egg, parsley, basil, cottage cheese, 1 ¼ cups mozzarella, and ¼ cup Parmesan.
Spread ¼ of the Alfredo mixture in a thin layer in the bottom of the prepared baking dish. Top with 4 no-boil lasagna noodles, ⅓ of the cottage cheese mixture, and another ¼ of the Alfredo mixture.
Repeat layers two more times. Finish with remaining 1 ¼ cups of mozzarella and ¼ cup of Parmesan cheese on top. Cover with foil sprayed with cooking spray.
Bake for 40 minutes. Remove foil, and bake for another 10 minutes (or until heated through and cheese on top is browned). Let stand for at least 15 minutes before slicing and serving.