Combine mirin and soy sauce in a medium saucepan; bring to a boil over medium-high heat. Cook until reduced to 3/4 cup (about 12 minutes). Pour into a bowl; cool completely. Stir in oil. (Mixture will thicken as it cools.)
Combine half of soy mixture and steaks in a shallow dish, turning to coat. Cover and marinate in refrigerator 2 hours, turning occasionally. Reserve remaining soy mixture.
Remove steaks from refrigerator; discard soy mixture in dish. Let steaks stand at room temperature 20 minutes.
Pat steaks lightly with a paper towel. Place steaks on grill rack coated with cooking spray; grill 3 minutes. Turn steaks over; spoon reserved soy mixture over tops of steaks. Grill 3 minutes or until desired degree of doneness. Remove from grill. Let stand 5 minutes before serving.