Bring a large pot of water to a boil for the egg noodles.
Heat a medium sauce pot over medium heat. Add extra-virgin olive oil, 1 turn of the pan, about 1 tablespoon, and 2 tablespoons butter. When butter melts, add in chopped button mushrooms and cook 5 minutes, until just tender. Sprinkle in a couple of tablespoons flour and cook 1 minute. Whisk in chicken stock and bring to a bubble then stir in the whole milk or cream. Reduce heat to low and gently simmer. Season the sauce with nutmeg, salt, and pepper.
Heat a nonstick skillet over medium-high heat. Add 2 tablespoons extra-virgin olive oil, 2 turns of the pan then the shallots and mushrooms. Cook the mushrooms 8 minutes, until tender. Season with thyme, salt, and pepper and deglaze the pan with 1/3 cup dry white wine or a little stock. Reduce heat to medium-low and let the liquid cook off.
Preheat broiler to high.
While the mushrooms cook, drop egg noodles in water, salt the water and cook noodles to al dente, with a bite to them. Drain noodles and return to hot pot. Add the sauce to the pot and toss noodles to coat in sauce.
Lightly cover a casserole dish with softened butter then transfer cream of mushroom noodles to the casserole dish and top with wild mushroom ragout and the shredded cheese. Place the casserole under the broiler and melt and bubble the cheese until brown at edges. Garnish with chives.