In a bowl, blend brown sugar, soy sauce, lemon juice, sesame oil, chile flakes, broth, and minced ginger; set sauce aside.
Cut slender Asian eggplant into 1- by 2-in. slices or regular globe into 1 1/2-in. chunks; or cut small, egg-shaped Indian eggplant in half lengthwise. Set aside.
Bring 3 cups water to a boil in a 12-in. wok or frying pan over high heat. Add broccolini; cook, covered, until stems are just tender, 3 to 4 minutes. Transfer to a bowl of ice water; when cool, drain and set aside.
Drain and dry wok, add canola oil, and heat over high heat. Add slivered ginger; cook, stirring, until golden, 2 minutes. Transfer ginger to paper towels. Pour oil into a bowl; return 1 tbsp. oil to wok.
Add half of eggplant to wok over high heat. Cook, turning often, until lightly browned, 3 minutes. Transfer to a bowl. Add remaining oil to wok with remaining eggplant and the shallots. Cook as before.
Return eggplant to wok with sauce. Bring to a boil; cover, reduce heat, and simmer, stirring occasionally, until eggplant is soft when pressed, 8 to 15 minutes.
Lay broccolini on eggplant, cover, and heat 2 minutes. If needed, cook, uncovered, over high heat until most of liquid evaporates, 1 minute. Stir in cilantro and mint and top with fried ginger.