Cut the eggplants into batons about ¾ inch thick and 2 ¾ inches long. Sprinkle with salt, mix well, and set aside for at least 30 minutes.
Rinse the eggplant, drain well, and pat dry with paper towels. Heat the deep-frying oil to around 390°F (200°C) (hot enough to sizzle vigorously around a test piece of eggplant). Add the eggplant, in 2 or 3 batches, and deep-fry for about 3 minutes, until tender and a little golden. Drain well on paper towels and set aside.
Carefully pour off all but 3 tablespoons oil from the wok and return to medium heat. Add the chile bean paste and stir-fry until the oil is red and fragrant; take care not to burn the paste (move the wok away from the burner if you think it might be overheating). Add the garlic and ginger and stir-fry until they smell delicious.
Tip in the stock or water, sugar, and soy sauce. Bring to a boil, then add the eggplant, nudging the batons gently into the sauce so they do not break apart. Simmer for a minute or so to allow the eggplant to absorb the flavors.
Give the potato starch mixture a stir and add it gradually, in about 3 stages, adding just enough to thicken the sauce to a luxurious gravy (you probably won’t need it all). Tip in the vinegar and all but 1 tablespoon of the scallion greens, then stir for a few seconds to fuse the flavors.
Turn out onto a serving dish, scatter over the remaining scallion greens, and serve.
