Preheat oven to 350°F (175°C). Prepare your ingredients: pull the rotisserie chicken meat, cook the egg noodles according to package directions, and sauté the sliced mushrooms until tender and browned.
Make the sauce: Melt butter in a small saucepan over medium heat. Whisk in the flour and cook for a few minutes until lightly golden and fragrant. Slowly whisk in the milk/cream and chicken broth, adding a little at a time to prevent lumps. Continue whisking until the sauce is smooth and thickened.
Season the sauce generously with salt. Taste and adjust seasonings as needed.
Assemble the tetrazzini: In a greased 9x13 inch baking dish, start by tossing the cooked egg noodles with about 1 cup of the prepared sauce. Layer this evenly in the pan.
Add the sautéed mushrooms over the noodles, followed by half of the remaining sauce. Then, spread the shredded chicken evenly on top, and cover with the rest of the sauce.
Sprinkle the 1/4 cup of Parmesan cheese evenly over the top of the entire dish. Bake for 30 minutes, or until the sauce is bubbly and the cheese is melted and golden.
For an extra browned and bubbly top, you can broil the tetrazzini for an additional 5 minutes, watching carefully to prevent burning. Let it rest for a few minutes before serving.