Cut the eggplants in 1/2 lengthwise and then slice crosswise into wedges, no more than 1-inch wide.
Heat a wok or large skillet over medium-high flame and add the oils; tilt the pan to coat all sides. When you see a slight smoke, add a layer of eggplant, stir-fry until seared and sticky, about 3 minutes. Season with salt and pepper. Remove the eggplant to a side platter and cook the remaining eggplant in same manner, adding more oil, if needed.
After all the eggplant is out of the pan, add the green onions, ginger, garlic, and chile; stir-fry for a minute until fragrant. Add the broth. In a small bowl, mix the soy sauce, vinegar, sugar, and cornstarch until the sugar and cornstarch are dissolved. Pour the soy sauce mixture into the wok and cook another minute, until the sauce has thickened. Put the eggplant back in the pan, tossing quickly, until the sauce is absorbed. Garnish with sesame seeds, Thai basil, and cilantro and serve.
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