In a small bowl, whisk together the stir-fry sauce ingredients. Set aside.
Wash the eggplants with the skin on and pat them dry with paper towels. Slice the eggplant lengthwise into quarters, then into 2-inch-long wedges, and transfer to a large mixing bowl. Add the cornstarch and toss until each piece is evenly coated. If any uncoated spots remain, add 1-2 tablespoons more cornstarch.
Heat 2 tablespoons vegetable oil in a large pan over medium-high heat until shimmering. Add the eggplant and cook until tender and golden brown while still holding its shape, gently turning occasionally, about 12-15 minutes. Remove and set aside on a wire rack or a paper towel-lined plate.
Reduce to medium heat and heat 1 tablespoon vegetable oil. Once hot, add the red and green chilies and stir-fry for 15 seconds. Avoid inhaling the fumes directly, as the chilies may cause coughing.
Re-whisk the stir-fry sauce, pour it into the pan, and simmer until thickened and glossy, about 30 seconds. Add the cooked eggplant and gently toss with the sauce for 30 seconds until evenly coated. Remove from the heat.
Sprinkle with toasted sesame seeds and serve.