Cut the eggplant lengthwise. Remove the two outer pieces, only using the 8 pieces. Sprinkle slices with salt and leave them in a colander for 20 minutes.
Pre-heat grill on high heat. Slice hindquarter into 8 slices. Pound to 1/4 inch. Season with salt and pepper. Brown venison on grill, approximately 30 seconds on each side. Remove and let rest.
Rinse eggplant in cold water. Pat dry.
Brush slices with oil and grill 4 minutes on each side.
Remove slices from grill. Place mozzarella slice, tomato slice, venison slice, and basil on top of the eggplant. Fold the eggplant over the filling. Place back on grill seam side down until cheese is melted.
Serve with balsamic vinegar and olive oil drizzled over the top.