Heat a medium skillet over medium heat. Add oil to pan; swirl to coat. Add onion, oregano, and garlic; sauté 3 minutes or until softened. Place onion mixture in a large bowl; cool slightly. Add chicken, Parmesan, 1/2 cup marinara, salt, and peppers to onion mixture. Mix well with hands. Shape into 8 (1/2-inch-thick) patties. Refrigerate 10 minutes.
Preheat broiler to high.
Heat a grill pan over medium-high heat. Coat pan with cooking spray. Add patties to pan; cook 10 minutes or until done, turning once.
Arrange bottom halves of rolls on a baking sheet. Place 1 patty on each roll. Top each with 1 tablespoon remaining marinara and 1 1/2 tablespoons cheese. Broil 90 seconds or until cheese melts. Top with basil and top halves of buns.