Heat the olive oil in the ovenproof skillet over medium-high heat. Add the mushrooms until soft and lightly browned, about 12 minutes. Add the chard and cook until limp and still bright, about 4 minutes, then reduce heat to medium.
In the medium bowl, whisk together the egg whites, eggs, salt, and pepper. Pour into the pan, making sure the mixture covers the pan completely. Sprinkle the chives and green onions around the top and gently pat them into the egg mixture. Cover and cook until almost set, about 3 to 4 minutes.
Sprinkle the cheese all over the top, then transfer to the oven and broil 3 minutes or until the egg is set. Remove from the oven and allow to cool for minute, then cut into wedges and serve.