Spread 1 tablespoon of the butter into a 13×9-inch baking dish.
In a large skillet over medium-high heat, melt 1 tablespoon of butter into 1 tablespoon of the oil.
Sprinkle chicken with salt and pepper, to taste. Add the chicken to the skillet, and cook until just cooked through, about 4 minutes each side. Transfer the chicken to a cutting board, and allow to cool. When cool enough to handle, shred the chicken into bite-sized pieces. Transfer the chicken to a large bowl.
Add 1 tablespoon of butter and 1 tablespoon of oil to the existing pan. Add the mushrooms, and saute until the mushrooms are golden, about 12 minutes. Add the onion, garlic and thyme. Saute until the onions are translucent, about 8 minutes. Add wine and simmer, about 2 minutes. Transfer the mushroom-onion mixture to the bowl containing the chicken.
Reduce the stovetop heat to medium-low. In the same skillet, melt 3 tablespoons butter. Add the flour. Whisk for 2 minutes. Whisk in the milk, cream, broth, nutmeg, the remaining 1 3/4 teaspoons of salt and the remaining 3/4 teaspoon of pepper. Increase the stovetop heat to high, cover the skillet and bring to a boil. Simmer the sauce uncovered, whisking often, until it thickens, about 10 minutes.
In a large pot over high heat, bring the salted water to a boil. Add the linguine. Cook the pasta until it is al dente, about 9 minutes. Drain the pasta.
Add the linguine, sauce, peas and parsley to the chicken mixture. Toss the chicken until the mixture is well blended. Transfer the pasta to the prepared baking dish.
In a small bowl, combine the cheese and breadcrumbs. Sprinkle the cheese mixture over the pasta. Dot the dish with the remaining 3 tablespoons of butter. Bake the dish uncovered, until the top is golden brown, about 25 minutes.
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