Cook broccoli florets in salted boiling water for 5 minutes, then sift out and reserve (don’t drain the water). Drop the egg noodles into the same boiling water and cook to al dente according to 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 in the sauce.
To prepare the casserole, heat a nonstick skillet over medium-high heat. Add the 2 tablespoons of EVOO, then the shallots. Cook for 1 minute and add the chicken to the skillet. 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 then transfer the cream-of-mushroom noodles to the dish and top with the chicken, then the cooked broccoli and finally the shredded cheese. Place under the broiler until cheese is melted and bubbly, and golden brown at the edges. Garnish with chives and serve.