For dough: In the bowl of a stand mixer, whisk together 2 cups (250 grams) flour, salt, and yeast by hand until combined. Add 1¼ cups (300 grams) water, oil, and egg; using the paddle attachment, beat at low speed until combined. With mixer on low speed, gradually add 2 cups (250 grams) flour, beating until a shaggy dough forms, 1 to 2 minutes.
Switch to the dough hook attachment. Beat at medium-low speed until a smooth, elastic dough forms, 12 to 15 minutes; add up to remaining ¼ cup (31 grams) flour,1 tablespoon (8 grams) at a time, if dough is too sticky.
Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 45 minutes.
Lightly punch down dough. Cover and let stand for 5 minutes.
Lightly spray a 13x9-inch baking pan with baking spray with flour.
For filling: In a medium bowl, whisk together oil, parsley, oregano, salt, garlic powder, and pepper.
Turn out dough onto a lightly floured surface, and roll into an 18x12-inch rectangle, with one long side closest to you. Brush oil mixture onto dough. Sprinkle mozzarella, provolone, and 2 ounces (57 grams) Parmesan onto oil, leaving a ½-inch border on one long side. Starting with long side opposite border, tightly roll up dough into a log, pinching seam to seal. Gently shape log to an even thickness. Using a sharp knife, cut log into 12 slices (about 1½ inches thick). Place, cut side up, into prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.
Preheat oven to 350°F (180°C).
In a small bowl, whisk together egg and 1 tablespoon (15 grams) water; gently brush onto rolls.
Bake until top is golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), 35 to 40 minutes. Sprinkle remaining 2 ounces (57 grams) Parmesan onto rolls, and bake until cheese is melted and lightly browned, 2 to 3 minutes more. Let cool for 10 to 15 minutes. Serve with marinara sauce. Garnish with parsley, if desired.