Melt the butter in a saucepan over medium heat. Whisk in the flour until smooth; cook, stirring, until the mixture lightens, about 1 minute. Remove from the heat; whisk in the milk, 2 teaspoons salt and the nutmeg. Return to medium heat and bring to a boil, whisking. Remove from the heat again; stir to cool slightly.
Heat a deep skillet over medium heat. Add the spinach and 1 teaspoon salt; cover and cook about 3 minutes. Toss with tongs, then continue to wilt, uncovered, about 2 more minutes. Drain and cool slightly, then squeeze the spinach until dry. Coarsely chop.
Gently reheat the sauce, then whisk in the whole egg and yolks; cook, stirring, until the sauce thickens. Stir in the spinach; heat through. Season with salt and pepper.