Preheat the oven to 400 degrees . Grease a 9-by-13-inch baking dish. Cook the pasta according to package directions until just barely al dente; drain.
Meanwhile, season the turkey with salt and pepper. In a large skillet, heat 1 tbsp. butter over medium-high heat. Add the turkey and cook until browned, 1 to 2 minutes. Transfer to a bowl.
Add 2 tbsp. butter to the skillet. Add the mushrooms and thyme; season with salt and pepper. Cook until the mushrooms start to soften, about 3 minutes. Transfer to the bowl with the turkey.
Melt the remaining 3 tbsp. butter in the skillet. Whisk in the flour and cook for 1 minute. Whisk in the milk and broth and bring to a boil, then simmer for 2 to 3 minutes to thicken. Stir in 1/2 cup parmesan and the peas and carrots. Season with salt and pepper.
Add the pasta and turkey-mushroom mixture to the sauce and toss well to combine. Transfer the pasta mixture to the prepared baking dish. Sprinkle with the remaining 1/2 cup parmesan. Bake until the top is golden-brown and the sauce is bubbling, 12 to 15 minutes (shorten the baking time by at least 5 minutes if youre planning on freezing this for later).