Preheat the oven to 450.
Use 1 tablespoon of the butter to grease a 13x9 inch baking dish.
In a large, nonstick frying pan, melt 1 tablespoon of the butter and the oil.
Sprinkle the chicken with 1/2 teaspoon each of salt and pepper. Add the chicken to the pan and cook until pale golden. Transfer the chicken to a plate to cool slightly.
In a large bowl, shred the chicken into bite size pieces.
Add the mushrooms to the same frying pan. Saute until the liquid evaporates and the mushrooms become pale golden, about 12 minutes. Transfer the mushroom mixture to the bowl with the chicken.
Melt the remaining 3 tablespoons butter in the same pan. Whisk in the milk, broth, cream, nutmeg, 1 3/4 teaspoon salt, and 3/4 teaspoon pepper. Increase the heat to high. Cover the sauce and bring to a boil. Then simmer, uncovered, whisking often, until the sauce thickens slightly, about 1 minute.
Bring a large pot of salted water to a boil. Add the linguine and cook. Drain.
Add the linguine, sauce, peas, and parsley to the chicken mixture. Toss until the sauce coats the pasta and the mixture is well blended.
Transfer the pasta to the prepared baking dish. In a small bowl, stir the cheese and bread crumbs to blend. Sprinkle over the casserole. Bake uncovered until golden brown on top - 25 minutes.