Preheat the oven to 400°. Grease a 9-by-13-inch baking dish and set to the side.
Cook the pasta according to package directions until just barely al dente; drain.
Add 2 tablespoons butter to the skillet. Add the mushrooms and thyme; season with salt and pepper. Cook until the mushrooms start to soften, about 3 minutes. During the last minute of cooking, add the chopped turkey, mix well, and transfer to a bowl and set to the side.
Melt the remaining 3 tablespoons 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 ½ cup Parmesan, peas and carrots mix, and turkey. Season with salt and pepper if desired.
Add the pasta and cooked mushrooms/turkey to the sauce. Toss well to combine. Transfer the pasta mixture to the prepared baking dish.
Sprinkle with the remaining ½ cup Parmesan. Bake until the top is golden-brown and the sauce is bubbling, 12 to 15 minutes. Carefully, remove from the oven and serve immediately.