Steam the spinach at a moderate heat for 3 to 4 minutes. Cool under running water, then drain, pressing out as much liquid as you can. Then chop finely. If using frozen spinach, defrost and squeeze dry.
Gently sauté the garlic and mushrooms in butter; turn up the heat and evaporate the juices.
When dry, add the spinach and transfer to a bowl. Cover and keep warm.
Measure 3 tablespoons of milk into a bowl. Bring the remaining milk to a boil. Stir the flour and egg yolk into the milk that is in the bowl, and blend well. Stir the boiling milk into the eggs and flour mixture; return to the pan and simmer to thicken.
Add the spinach mixture; season to taste with salt, pepper. and nutmeg.
Butter a 3 3/4 cup soufflé dish, paying special attention to the sides.
Sprinkle with a little of the cheese. Set aside.
Whisk the egg whites until they hold soft peaks.
Bring the spinach mixture back to a boil.
Stir in a spoonful of beaten egg whites, and then fold the remaining mixture into the remaining egg whites.
Turn into the soufflé dish; spread level. Sprinkle with the remaining cheese and bake in the oven for about 25 minutes until puffed, risen, and golden.