Prepare a charcoal grill with hot coals or heat a gas grill.
In a non-corrosive saucepan, reduce the sherry wine vinegar and sugar by boiling rapidly until reduced to 1/4 cup, about 5 minutes. It should be thick and syrupy. (You've just made your own balsamic vinegar.)
Meanwhile, heat a tablespoon of olive oil in a small saute pan, add the garlic and cook for 1 minute until browned.
Slice the eggplants vertically in 12 very thin slices with the skin on.
Brush the eggplant slices with olive oil and grill them until they are just about to burn, the duration will depend on how hot the grill is. You want some char here, nearly black is good. (You can also do this on a skillet.)
Drizzle the vinegar syrup over the eggplant slices and garnish with the garlic.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.