Wash the spinach and cut off the stems. Spin the excess water out. Drop the spinach into a large pot on the stove to wilt, about 3 minutes. (There will be enough water still clinging to the leaves to steam it.) When the spinach is cooked, drain, and plunge it into an ice-bath to seize the color. Remove the spinach and squeeze dry in a tea towel. Chop the spinach.
Heat half the butter in a frying pan, and gently saute the shallot. Add the spinach, and season with salt, pepper, and nutmeg. When hot, turn off the heat, and set aside.
Put the yolks in a metal bowl and set over a pot of gently simmering water (make sure the water does not touch the bottom of the bowl.) Season the eggs with salt, and pepper, and whisk until thick and frothy. Remove the bowl from the heat. In a separate bowl, beat the whites to stiff peaks, and fold into the yolk mixture.
Heat the remaining butter in a large frying pan for the omelette.
Cook's Note: a no-stick pan will simplify your life.
Pour the egg into the pan and cook until the bottom is set, and golden, and the top is still fluffy and soft, about 3 to 5 minutes. Scatter the spinach mixture all over the omelette, then the cheese. Cover the pan for a few minutes to soften the cheese. Uncover, and fold the omelette in half, and slide it onto a plate. Serve immediately.