Heat a large nonstick skillet coated with cooking spray over medium-high heat.
Add white onion; saut 7 minutes or until golden.
Add green onions and garlic, and saut 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.
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.
