Spread 1 tbsp of butter over a 13 by 9 by 2-inch baking dish.
Melt 1 tbsp each of butter and oil in a deep large nonstick frying pan over medium-high heat.
Sprinkle the chicken with 1/2 teaspoon each of salt and pepper.
Add the chicken to the hot pan and cook until pale golden and just cooked through, about 4 minutes per side.
Transfer the chicken to a plate to cool slightly.
Coarsely shred the chicken into bite-size pieces and into a large bowl.
Meanwhile, add 1 tbsp each of butter and oil to the same pan.
Add the mushrooms and saute over medium-high heat until the liquid from the mushrooms evaporates and the mushrooms become pale golden, about 12 minutes.
Add the onion, garlic, and thyme and saute until the onion is translucent, about 8 minutes.
Simmer until it evaporates, about 2 minutes.
Transfer the mushroom mixture to the bowl with the chicken.
Melt 3 more tbsp butter in the same pan over medium-low heat.
Add the flour and whisk for 2 minutes.
Whisk in the milk, cream, chicken stock, nutmeg, remaining 1 3/4 tsp salt and remaining 3/4 tsp pepper.
Increase the heat to high.
Cover and bring to a boil.
Simmer, uncovered, until the sauce thickens slightly, whisking often, about 10 minutes.
Bring a large pot of salted water to a boil.
Add the pasta and cook until it is tender but still firm to the bite, stirring occasionally, about 9 minutes.
Drain. Add the pasta, sauce, peas and parsley to the chicken mixture.
Toss until the sauce coats the pasta and the mixture is well blended.
Transfer the pasta mixture to the prepared baking dish.
Stir the cheese and breadcrumbs in a small bowl to blend.
Sprinkle the cheese mixture over the pasta.
Dot with the remaining 3 tbsp of butter.
Bake, uncovered, until golden brown on top and the sauce bubbles, about 25 minutes.