Place a slice of eggplant on each plank (2 slices if using long planks). Top each slice with a slice of burrata (spoon on some of the cream as well), followed by a generous dollop of tomato sauce.
Sprinkle with Parmigiano-Reggiano and a tuft of slivered basil, if using. Build the second layer of eggplant, burrata, tomato sauce, Parmigiano-Reggiano, and basil. Crown with a slice of eggplant.
If you like a crisp top, sprinkle the top eggplant slice with breadcrumbs and drizzle with extra virgin olive oil. The eggplant Parmesans can be assembled and refrigerated several hours ahead of time and grilled at the last minute.
Return the assembled eggplant Parmesans on their planks to the grill, placing over indirect heat. Grill until the tomato sauce is bubbling, the cheese is melted, and the tops are browned, 10 to 20 minutes.
Serve immediately, preferably still on the cedar planks.