To make the sauce: In a medium saucepan, heat the sesame oil over medium heat until hot, but not smoking. Add the garlic and ginger and saute stirring occasionally for 3 minutes. Add the lime juice, ketchup, brown sugar, and chile peppers and bring to a boil, stirring occasionally. Reduce heat to low and simmer, stirring once in a while, for 20 minutes or until the sauce is thick enough to coat the back of a spoon. Remove from the heat and set aside.
Put the eggplant rounds into a medium bowl along with the vegetable oil and salt and pepper to taste, and toss to coat. Cook the eggplant on the grill over medium-high heat and cook until browned and soft, about 4 minutes per side. To check for doneness: Cut into the eggplant and check to be sure the interior looks moist. Brush the eggplant on both sides with the chile sauce, leave it on the grill for about 10 seconds and then remove it from the grill. Sprinkle with the scallions, coriander and red pepper flakes. Serve warm.