1Brush grill rack with vegetable oil. Heat coals or gas grill for direct heat. With kitchen scissors, cut each hen in half along backbone from tail to neck.
2Brush hen halves with basil-flavored oil. Sprinkle with salt and pepper. Insert barbecue meat thermometer so tip is in thickest part of inside thigh muscle and does not touch bone.
3Place hen halves, skin side down, on grill rack. Cover and grill over medium heat 30 to 40 minutes, brushing occasionally with basil-flavored oil and turning once, until juice of hens is clear when thickest pieces are cut to bone (at least 165°F). Let stand loosely covered about 15 minutes before serving.
4While hens are grilling, make sauce. In 1-quart saucepan, heat all sauce ingredients except basil to boiling; reduce heat. Simmer uncovered about 25 minutes or until sauce thickens slightly. Stir in basil. Simmer 5 minutes. Serve warm with hens.