Preheat oven to 350°F. Spray a 13 x 9-inch baking dish with nonstick cooking spray; set aside.
In a large skillet (about 12-14 inches) or in a Dutch oven, melt butter over medium heat. Add the mushrooms, onion, and garlic; cook until tender, about 8 minutes. Season with a pinch of salt and some ground black pepper to taste. Whisk in the flour; cook, whisking constantly, for 1 minute. Gradually whisk in the broth, wine, and cream; bring to a gentle boil.
Reduce the heat to low; stir in the Parmesan cheese, 1 teaspoon of kosher salt, poultry seasoning, and ¼ teaspoon of black pepper. Cook, stirring regularly, until the cheese melts and the sauce thickens slightly, about 5-7 minutes. Stir in the chicken, rice, peas, and parsley. Taste and season with additional salt and pepper, if necessary. Transfer the mixture to the prepared dish.
In a small bowl, combine the cracker crumbs with the melted butter. Sprinkle over the top of the casserole.
Bake, uncovered, for 25-30 minutes, until the filling is hot and bubbly and the top is golden brown. Let stand for 5-10 minutes before serving.