Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Peel the eggplants lengthwise in 1-inch intervals to create stripes (this helps with oiliness) then slice each eggplant crosswise into 1/2-inch-thick rounds. Add the eggplant to a large bowl with the olive oil. Sprinkle in the oregano and toss to combine. Season well with salt and cracked black pepper.
Place the seasoned eggplant slices over direct heat and cook until nicely charred and beginning to soften, 2 to 3 minutes per side. Move the slices over indirect heat to finish cooking as you add more pieces to direct heat. Top each piece with a slice of tomato and a slice of mozzarella. Close the grill and allow the cheese to melt and the tomato to warm through.
Remove the eggplant to a baking sheet and grate Parmesan all over the top. Place the baking sheet back on the grill over direct heat to warm up and melt the Parmesan. Transfer to a platter and top with torn basil leaves and a drizzle of olive oil. Serve!