Heat 1 tablespoon of the oil in an ovenproof 10 inch skillet. Add onion. Cook over low heat, stirring occasionally, until softened but not colored, about 15 minutes. Add the chard and mushrooms. Continue cooking, stirring occasionally, until all the moisture has cooked off and chard is tender, about 15 minutes.

In a large bowl, stir together eggs, garlic, thyme, salt, and pepper. Stir in chard mixture, Gruyere and half the Parmesan, until blended. Heat oven to 400°F. Wipe out the skillet,then heat remaining tablespoon oil over medium-high heat for 2 minutes. Pour in egg mixture; cook 1 minute, stirring once.

Reduce heat to low and cook until eggs are set but still a little moist on top, 10 minutes longer. Sprinkle with remaining tablespoon of Parmesan. Place skillet in oven. Bake 10 minutes or until cooked through and lightly browned.