Preheat oven to 400°F. Season chicken breasts with salt and pepper; place them in an even layer on a greased baking sheet. Bake for 30-40 minutes, until completely opaque all the way through and registers 165°F on an instant-read thermometer.
In a mixing bowl, mix ricotta cheese, blue cheese, celery, parsley, and egg, set aside.
Cook noodles according to package directions; drain and drizzle with olive oil to prevent sticking.
Let chicken cool slightly before shredding it. Reduce oven temperature to 350°F.
Spread a little of the chicken mixture on the bottom of a greased 13×9-inch casserole dish. Spread 4 noodles in an even layer over that, followed by a third of the ricotta mixture, then 1 cup mozzarella cheese, and ⅔ cup cheddar cheese. Repeat layers twice.
Cover with foil and bake for 20 minutes. Uncover; bake 25 minutes longer or until cheese is bubbly. Let cool for 10-15 minutes before slicing and serving with scallions.
