Heat your barbecue over high heat, brush the eggplants with a little oil and season with salt. Grill the eggplants for about 8 minutes, turning once or twice, until softened and browned.
While the eggplant is cooking, place a small saucepan on the barbecue (or over medium heat on your stove) and add the remaining olive oil and the garlic. Cook the garlic for about 3-5 minutes until just starting to brown. Remove from the heat and stir through the sherry vinegar and parsley.
Arrange the eggplant on a serving platter, spoon over the garlic oil, then dust with the smoked paprika.
