Preheat oven to 400°F (204°C). Butter a 13x9x3-inch baking dish with 1 tablespoon of the butter or spray generously with a nonstick cooking spray; set aside.
In an 8-quart stockpot, add 1 tablespoon of butter and 1 tablespoon of olive oil. Stir in mushrooms (1 pound) and cook over medium-high heat until the mushrooms are golden brown and the liquid has evaporated.
Stir in the onion (1), garlic (1 tablespoon), and thyme (1 tablespoon) and continue to cook until the onions are translucent about 6-8 minutes. Stir in wine (1/2 cup) and cook until it evaporates about 2 minutes. Transfer the mushroom mixture to the bowl.
Add 3 more tablespoons of the remaining butter to the same stockpot with the heat on medium-low. Sprinkle the flour (1/3 cup) over melted butter and whisk constantly for 1-2 minutes to remove flour taste. Remove from heat and slowly whisk in milk (4 cups), heavy cream (1 cup), and broth (1 cup) until smooth.
Stir in nutmeg (1/8 teaspoon), remaining 2 teaspoons salt, and remaining 3/4 teaspoons black pepper. Turn up the heat to high and bring to a boil. Immediately reduce to a simmer and continue to cook, whisking often, until the sauce thickens, about 9-11 minutes.
Cook the linguine (12 ounces) according to package instructions, in well-salted water (2 tablespoons salt), until al dente. Drain well.
Add the cooked chicken (4 cups) and onion mixture and drained linguine to the sauce along with the peas (3/4 cup), and parsley (1/4 cup). Mix well until the sauce coats the pasta and the mixture is well blended.
Place the baking dish on a rimmed baking sheet. Pour mixture into prepared baking dish. Top with remaining cheese and dot with butter and bake in the preheated oven, uncovered, for about 25 minutes or until heated through and golden brown on the top.
Sprinkle top with parsley (2-3 tablespoons) to garnish and serve.