Combine 2 cups flour, sugar, 1 3/4 teaspoons salt, and yeast; stir. Add water and 2 tablespoons oil, beating at low speed of an electric mixer until blended. Beat 2 minutes at medium speed. Stir in enough of the remaining 3 1/2 cups flour to make a soft dough.
Sprinkle 1/4 cup flour over work surface. Turn dough out; knead until smooth and elastic (about 10 minutes). Place in a bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled.
Position knife blade in food processor bowl, and add parsley, onions, and garlic. Process 20 seconds. Add remaining 1/2 teaspoon salt, 2 tablespoons oil, and pepper; process 15 seconds.
Punch dough down; divide in half. Sprinkle 1 tablespoon flour over work surface. Roll 1 portion into a 16- x 10-inch rectangle. Spread half of parsley mixture over dough, leaving a 1-inch border; sprinkle with 1/4 cup cheese. Roll up dough, starting at long side. Pinch seam and ends to seal. Repeat procedure.
Place loaves, seam side down, on a baking sheet coated with cooking spray. Make 1/4-inch-deep slits in each loaf. Cover and let rise in a warm place, free from drafts, 35 minutes or until doubled. Brush with egg white. Bake at 375° for 30 minutes or until loaves sound hollow when tapped.