Preheat oven to 350°.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add white onion; sauté 7 minutes or until golden. Add green onions and garlic, and sauté 1 minute. Stir in chard; cook 2 minutes or until chard wilts. Stir in parsley and mint, and cook 1 minute. Place in a large bowl; cool slightly. Stir in cheeses, salt, pepper, and egg whites.
Place 1 phyllo sheet on a large cutting board (cover remaining phyllo to prevent drying), and coat with cooking spray. Top with 1 phyllo sheet, and coat with cooking spray. Repeat procedure with 3 additional sheets.
Cut phyllo stack into a 14-inch square. Place square in center of a 13 x 9-inch baking dish coated with cooking spray, allowing phyllo to extend up long sides of dish. Cut 14 x 4-inch piece into 2 (7 x 4-inch) rectangles. Fold each rectangle in half lengthwise. Place a rectangle against each short side of dish. Spread the chard mixture evenly over phyllo.
Place 1 phyllo sheet on a large cutting board (cover remaining phyllo to prevent drying), and coat with cooking spray. Top with 1 phyllo sheet, and coat with cooking spray. Repeat procedure with remaining phyllo sheets. Place 18 x 14-inch phyllo stack over chard mixture. Fold phyllo edges into center. Coat with cooking spray. Score phyllo by making 2 lengthwise cuts and 3 crosswise cuts to form 12 rectangles. Bake at 350° for 40 minutes or until golden.
Note: Cut the phyllo stacks so they fit in and up the long side of the baking dish. Arrange folded section against short edges of dish to encase filling.
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