Position a rack in the upper third of the oven and preheat to 425 degrees F. Heat the olive oil in a deep ovenproof skillet over medium-high heat. Add the mushrooms, season with salt and pepper and cook, stirring occasionally, until lightly browned, about 4 minutes. Transfer to a plate and wipe out the skillet.
Melt the butter in the skillet over medium heat; add the flour and cook, whisking, 3 minutes. Whisk in the milk and chicken broth until smooth; simmer, whisking constantly, until slightly thickened, about 5 minutes. Whisk in 1/2 teaspoon salt and the nutmeg.
Add the chicken, mushrooms, gnocchi and spinach to the sauce and stir until coated and the spinach wilts. Sprinkle with the parmesan, transfer the skillet to the oven and bake until bubbling, about 20 minutes. Turn on the broiler; broil until lightly browned on top, about 3 more minutes.