Step 1

Preheat the oven to 450°F. Spread 1 tablespoon of butter over a 13x9x2-inch baking dish. Melt 1 tablespoon of butter and the oil in a deep large nonstick frying pan over medium-high heat. Sprinkle the chicken with 1/2 teaspoon of each 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.

Step 2

Meanwhile, add the mushrooms to the same pan. Sauté over medium-high heat until the liquid from the mushrooms evaporates and the mushrooms become pale golden, about 12 minutes. Add 1 tablespoon of butter. Add the onion, garlic, and thyme, and sauté until the onion is translucent, about 8 minutes. Add the wine and simmer until it evaporates, about 2 minutes. Transfer the mushroom mixture to the bowl with the chicken.

Step 3

Melt the remaining 3 tablespoons butter in the same pan over medium-low heat. Add the flour and whisk for 2 minutes. Whisk in the milk, broth, cream, nutmeg, remaining 1 3/4 teaspoon salt, and remaining 3/4 teaspoon pepper. Increase the heat to high. Cover and bring to a boil. Simmer uncovered until the sauce thickens slightly, whisking often, about 1 minute.

Step 4

Bring a large pot of salted water to a boil. Add the linguine and cook until it is tender but still firm to the bite, stirring occasionally, about 9 minutes. 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.

Step 5

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. Bake uncovered until golden brown on top and the sauce bubbles, about 25 minutes.