In medium bowl, smash blackberries with a fork until slightly chunky.
Brush 1 side of bread slices with oil; place oiled side down on work surface. Spread 4 bread slices with smashed blackberries; top with cheese, basil and remaining 4 bread slices, oiled side up.
Heat griddle or grill pan over medium-low heat. In batches if necessary, cook sandwiches 10 minutes or until bread is golden brown and cheese melts, turning once and pressing down occasionally with large spatula. Cut sandwiches in half to serve.
Percent Daily Values are based on a 2,000 calorie diet. Your daily values may be higher or lower depending on your calorie needs.