To make the meat, if using flank steak, line a cutting board with plastic wrap. Place 1 piece of meat on plastic wrap and cover it with a second piece of plastic wrap. Pound out steak to 3/8” even thickness. Place on a rimmed sheet pan.
Season both sides of beef with salt and pepper. On one side, sprinkle with chile flakes (if using), garlic, parsley, pecorino, and lay slices of prosciutto atop cheese. Add eggs and mozzarella. Roll meat tightly into rolls. Secure meat with toothpicks. Repeat with remaining meat rolls. Set aside.
To make the sauce, in a pot over medium-high heat, add 4 tablespoons oil. When oil shimmers, add onions and garlic and cook until softened and beginning to brown, about 5 minutes. Move onions and garlic to pan perimeter. Add remaining 1 tablespoon oil in middle of pan, add oregano and basil and cook until fragrant, 20 seconds. Add tomato paste and cook with spices, until color deepens, 3 minutes.
Combine herb-tomato paste and onion-garlic mixture together. Add diced tomatoes, tomato sauce, chicken broth, and soy sauce and stir to combine. Season with salt and pepper. Bring mixture to a boil. Cover, reduce heat to low, and simmer, about 5 minutes.
Meanwhile, in a large frying pan over medium-high heat, add olive oil. When oil shimmers, add garlic and cook, about 1 minute. Add meat rolls and cook until browned on the first side. Turn to second side and cook until browned. Continue to turn until all sides are browned.
Transfer meat to tomato sauce, submerging meat into sauce. Partially cover and let simmer until internal temperature reaches 140°, about 45 minutes.
Transfer meat rolls to a plate, remove toothpicks. Add some sauce to a serving platter, place rolls of meat over sauce. Spoon out more sauce over involtini. Garnish with pecorino Romano cheese and parsley.
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