In a skillet set over medium heat, cook spinach covered, with the water left clinging on the leaves. Drain and transfer spinach to ice bath. Squeeze out all liquid. Coarsely chop.
In a saucepan set over medium heat, melt the butter. Add the flour and cook, whisking for 2 to 3 minutes. Add stock in a stream, whisking until smooth. Add the mushrooms. Cook until tender. Add milk and stir until incorporated. Season with nutmeg, salt and pepper. Add spinach and stir to coat with sauce. Let cook for a few minutes, stirring.