In a bowl, combine eggplant, soy sauce, apple cider vinegar, sesame oil, brown sugar, ginger, and garlic, toss to coat well. Marinate the eggplant for 10-15 minutes.
Heat the skillet with oil. Add the marinated eggplant, cook until golden brown. Remove from the pan and set aside. To make the sauce, combine the leftover marinade with cornstarch. Stir until well incorporated and until the cornstarch dissolved.
In the same skillet, add the carrot, saute for a minute, add the eggplant, stir to combine.
Add the sauce, stir, cook until the sauce thickened. Sprinkle green onions and sesame seeds on top. Serve with steamed rice.