Heat 2 tablespoons olive oil in a large skillet over medium heat. Add onion; cook and stir until tender, about 10 minutes. Season with 1 teaspoon salt and 1 teaspoon pepper.

Combine parsley, Parmesan cheese, garlic, hard-boiled egg, 1 pinch salt, and 1 pinch black pepper in a bowl; drizzle in about 1 tablespoon olive oil, stirring continuously, until a sticky consistency is reached.

Spread parsley filling in a thin layer on flank steak. Roll steak in a jelly-roll style; fasten closed with toothpicks to contain filling.

Add remaining 1 tablespoon olive oil to skillet with onion mixture over medium heat, then add steak roll, turning until browned on all sides, 2 to 3 minutes per side. Add tomato sauce, chopped tomato, and sugar; simmer over very low heat until steak cooked through and sauce slightly reduced, 1 to 1 1/2 hours.