Preheat oven to 400. Bring a large pot of salted water to a boil (for pasta). In a large saucepan, melt 2 tablespoons butter over high heat. Add mushrooms, and season with salt and pepper. Cook, tossing frequently, until tender and browned, 8 to 10 minutes. Transfer to a bowl, and set aside.
Make sauce: In same saucepan, melt remaining 4 tablespoons butter over medium heat. Add flour; cook, whisking, about 1 minute. Whisking constantly, gradually add milk, broth, and wine. Bring to a boil; reduce to a simmer, and add 2 cups Parmesan and thyme. Season with salt and pepper.
Cook pasta 2 minutes less than package instructions for al dente; drain and return to pot. Add sauce, chicken, peas, and mushrooms. Toss well to combine. Divide between two shallow 2-quart baking dishes; sprinkle with remaining Parmesan. Freeze (see below) or bake until browned, about 30 minutes. Let stand 10 minutes before serving.
To freeze: After placing pasta mixture in baking dishes and sprinkling with Parmesan (step 3), cool to room temperature. Cover tightly with aluminum foil, and freeze up to 3 months.
To bake from frozen: Bake, covered with foil, at 400 degrees, until center is warm, about 2 hours. Uncover, and bake until top is browned, about 20 minutes more. Serve.
To bake from thawed: Thaw overnight in refrigerator. Bake, covered with foil, at 400 degrees, until center is warm, about 30 minutes. Uncover, and bake until top is browned, about 20 minutes.