Preheat the oven to 375 degrees and coat a shallow 1 1/2- to 2-quart baking dish with cooking spray.
In a large pot of boiling water, cook the noodles according to package directions, until al dente.
Meanwhile, in a large skillet over medium heat, melt the butter, then add flour, stirring, for 1 to 2 minutes. Gradually pour in broth and cook, stirring, until it thickens, about 2 to 3 minutes. Season with salt and pepper. Stir in milk and cook until very hot and smooth, another 3 minutes.
Put the turkey in a bowl and add half the sauce and stir the other half into the cooked noodles. Spoon the noodles into the prepared baking dish. Top with the turkey and spread evenly.
Bake, covered, for 15 minutes. Continue to bake, uncovered, for another 15 minutes, or until the top is golden brown and crisp.