Pour about 3½ liters of water into a large pot; add the coarse salt; cover and bring to a boil.
Meanwhile, carefully sort through the spinach to remove any wilted leaves; trim the stems; rinse the spinach thoroughly under running water, then add it to the prepared boiling water and cook uncovered for 15 minutes.
Preheat the oven to 200°C.
Prepare the béchamel sauce: heat the milk in a saucepan.
Melt the butter in another saucepan—preferably one with a heavy bottom; sprinkle in the flour; cook for 2 to 3 minutes over low heat, stirring constantly with a wooden spoon; then gradually pour in the hot milk and let the sauce thicken, stirring constantly.
When the sauce has reached the desired consistency, add the grated Gruyère; season generously with salt and pepper; add a pinch of grated nutmeg; stir to combine.
Once the spinach is cooked, drain it by squeezing it firmly to remove as much water as possible; then chop it.
Separate the eggs, placing the whites in a bowl; add the yolks to the béchamel sauce, stirring well with a wooden spoon; mix the béchamel sauce into the chopped spinach.
Add a pinch of fine salt to the egg whites and beat them with a whisk until stiff peaks form.
Add the egg whites to the béchamel-spinach mixture, folding the mixture very gently to incorporate air and taking care not to beat it.
Grease a soufflé dish; pour the mixture into it; it should fill the dish two-thirds of the way up.
Heat the bottom of the pan briefly, then place it in the oven on the lowest rack.
Bake for 40 minutes; serve the soufflé straight from the oven, in the baking dish.
Advice
The precautions to take with a spinach soufflé are the same as with any other soufflé: make a very thick and smooth béchamel sauce; beat the egg whites until stiff peaks form; do not open the oven door while the soufflé is baking; the temperature must remain constant, otherwise the soufflé will collapse. Increase the oven temperature 7 to 8 minutes before the soufflé is done baking.