Bring ½ inch water to a boil in a 10- to 12-inch ovenproof heavy skillet (not cast-iron), then add half of spinach and cook, turning with tongs, until wilted, about 30 seconds. Add remaining spinach and wilt in same manner, then cook, covered, over moderately high heat until spinach is tender, about 2 minutes. Drain in a colander and cool under cold running water. Gently squeeze handfuls of spinach to remove as much liquid as possible, then coarsely chop.

Wipe skillet dry, then cook onion and garlic in butter over moderately low heat, stirring, until softened, 2 to 3 minutes. Add mushrooms and increase heat to moderate, then cook, stirring, until mushrooms are softened and have exuded liquid, about 3 minutes. Stir in cream, salt, pepper, nutmeg, and chopped spinach and bring to a simmer** (see note). Remove skillet from heat and make 4 large indentations in spinach mixture. Break an egg into each indentation and bake, uncovered, until egg whites are set but yolks are still runny, 7 to 10 minutes. Lightly season eggs with salt and pepper, then sprinkle with cheese.