Put the wood chips in a deep bowl and pour in hot water to cover them. Keep the chips submerged with a small plate or saucer.
If you are using a charcoal grill, light the coals. When the flames have died down, but the embers are glowing brightly, use a long-handled tool to separate the coals into two piles. Drain the wood chips and sprinkle them over the coals. When they start smoking, add the eggplant slices to the parts of the grill over the coals and cook, covered, 10–15 minutes, flipping once. As the slices become cooked, move them to the center where they will still get smoky, but not char.If you’re using a gas grill, preheat it to medium. Drain the wood chips and wrap them in a tightly sealed packet of two layers of aluminum foil. With the tip of a knife, make several holes in the top of the packet, then place it over the flames, under the grill’s rack. When it starts to smoke, add the eggplant slices; cover and cook 10 to 15 minutes, flipping once. As the slices become cooked, move them to a cooler part of the grill or an upper rack where they will still get smoky, but not char.In a small saucepan, reheat the mole over medium-low heat.
On a plate, put one smoked eggplant disk, top with a slice of cheese, and repeat 2 or 3 more times, ending with an eggplant disc. Transfer the stack to a baking sheet. Repeat for the remaining stacks. Place the baking sheet on the surface of the grill and cook, covered, for just a few minutes, until the cheese melts.
Top the stacks with the hot mole and serve immediately.