In a medium bowl, stir the yogurt, cucumber, mint and garlic; season with salt and pepper.
Place a sheet of heavy-duty aluminum foil about 5 inches longer than the fish on a work surface. Arrange half the orange slices in the center of the foil. Place the fish on top, skin side down; season generously. Top with the remaining orange slices and the shallot and basil. Drizzle with the EVOO and vinegar. Top with another large foil sheet; crimp the edges to seal tightly. Repeat with two more large foil sheets, creating a double-layer packet. Refrigerate or pack over ice.
Heat a grill to medium-high. Add the foil packet and grill until a toothpick inserted through the foil into the fish meets no resistance, about 12 minutes. (Don't push the toothpick through the bottom layer of foil or the juices will run out!) Serve with the yogurt sauce.