Cook pasta according to package directions, omitting salt and fat.
While pasta cooks, combine oil, garlic, tomatoes, and onion on prepared pan; toss. Broil 4 to 6 minutes. Transfer mixture and any liquid from pan to a blender. Add 2 tablespoons basil and tomato paste; secure blender lid on blender. Remove center piece of blender lid; cover with a kitchen towel. Blend until smooth.
Drain pasta; return to pan. Stir in tomato sauce, salt, pepper, and chicken. Cook over medium heat until heated. Place spaghetti mixture on a serving platter. Chop remaining 2 tablespoons basil leaves. Sprinkle basil and Parmesan evenly over spaghetti.