Preheat oven to 350°. Remove and discard ribs from Swiss chard. Rinse with cold water; drain and coarsely chop.
Whisk together eggs and next 4 ingredients in a large bowl; stir in bread and half of cheese.
Saute mushrooms and next 3 ingredients in hot oil in a large skillet over medium-high heat 8 minutes. Stir in Swiss chard, and sauté 2 minutes. Fold vegetable mixture into egg mixture. Pour into a lightly greased 11- x 7-inch baking dish. Sprinkle with remaining cheese.
Bake at 350° for 35 to 40 minutes or until center is set. Let stand 5 minutes.