Heat a gas grill to medium high or prepare a medium-hot charcoal fire.
Using a serrated knife or a mandoline, slice the eggplant into rounds as thin as possible, about 1/4 to 1/8 inch. Spread the sliced eggplant (you may have to overlap them slightly) on two large baking sheets. Brush the tops with 1/2 cup of the oil and season with salt and pepper.
Smash the garlic and a pinch of salt into a paste following the directions here. In a small bowl, whisk together the remaining 1/4 cup oil and the garlic paste. Whisk in the lemon juice or vinegar and season with salt and pepper.
Once the fire is hot, grill both sides of the eggplant (in batches if necessary) until softened and cooked through, about 2 minutes per side. It’s fine if it chars slightly.
Transfer the eggplant to a medium serving bowl, add the garlic oil, and toss gently so as not to tear the eggplant. Let marinate for at least 15 minutes and up to 2 hours at room temperature (or refrigerate for up to 24 hours, but bring to room temperature before serving). Toss in the mint just before serving.