Rinse chard and place in a large pot (leaves should be wet). Cover and cook over medium heat until tender. 5 to 10 minutes. Drain. When cool enough to handle, squeeze out excess water.
Preheat broiler. While it’s preheating, combine eggs, milk, Parmesan, nutmeg, salt and pepper in a large bowl. Stir in spaghetti and chard.
Melt butter in a large ovenproof skillet on the stove over low heat. Add egg mixture; cover and cook until top is almost set, 10 to 13 minutes.
Sprinkle frittata with mozzarella and broil 3 minutes, or until golden brown.