Bring a large pot of water to a boil.
Pre-heat the broiler.
Make the mushroom sauce: heat a medium size saucepot over medium heat. Add the EVOO and the butter. When the butter melts, add the mushrooms and cook for 5 minutes, until just tender. Sprinkle in the flour and cook for 1 minute. Whisk in the chicken stock, then bring up to a bubble and stir in the milk or cream. Reduce the heat to low and gently simmer. Season the sauce with nutmeg, salt and pepper.
When the water is boiling, add some salt and the broccoli florets. Cook for 5 minutes, then lift out the broccoli with a slotted spoon and reserve (don’t drain the water). Drop the egg noodles into the same boiling water and cook to al dente, according to the package directions. Drain the noodles and return them to the hot pot. Add the creamy sauce to the pot and toss the noodles to coat.
To prepare the casserole, heat a nonstick skillet over medium-high heat with 2 tablespoons of EVOO. Add the shallots, cook for 1 minute, then add the chicken. Cook until lightly browned all over and firm, 6-7 minutes. Season with the thyme, salt and pepper. Deglaze the pan with the wine or stock, reduce the heat to medium-low and let the liquid cook off.
To assemble, lightly coat a casserole dish with butter. Transfer the cream-of-mushroom noodles to the dish and top with the chicken, followed by the cooked broccoli and finally, the shredded cheese. Place under the broiler until the cheese is melted and bubbly and golden brown at the edges. Garnish with chives and serve with a green salad alongside.