Bring 2 quarts water to a boil in a large Dutch oven. Add the chard and spinach; cook 2 minutes or until tender. Drain well. Place chard mixture on several layers of paper towels; squeeze until barely moist.
Preheat oven to 375°.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add onion; sauté 5 minutes or until tender. Add garlic; sauté 1 minute. Add chard mixture, dill, and parsley, stirring well to combine. Cook 1 minute or until thoroughly heated. Remove from heat. Combine chard mixture, cheese, eggs, and egg whites, tossing well to combine. Stir in pepper and salt.
Place 1 phyllo sheet on a large cutting board or work surface (cover remaining dough to prevent drying); lightly coat phyllo sheet with cooking spray. Place phyllo sheet in a 9-inch pie plate coated with cooking spray, allowing edges to overlap plate rim. Repeat the procedure with 6 additional phyllo sheets, placing sheets in a crisscross design. Spoon the spinach mixture over phyllo. Lightly coat each of remaining 3 phyllo sheets with cooking spray, and place sheets over spinach mixture in a crisscross design. Roll excess phyllo into the dish to create a decorative edge; press lightly to hold. Cut 4 (2-inch) slits in top of pie; cover with foil. Bake at 375° for 10 minutes. Uncover and bake an additional 30 minutes or until crust is crisp and golden. Cut pie into 8 wedges.