How to Make It
Rinse rice until water runs clear; drain. Transfer to a saucepan and add 1½ cups water. Bring to a boil over high; reduce heat to medium and simmer, covered, until liquid is absorbed and rice is tender, about 15 minutes. Remove from heat; let stand, covered, for 3 minutes. Fluff with a fork.
While rice cooks, heat 1 tablespoon oil in a large, high-sided skillet over medium-high. Add cumin, mustard, and coriander; cook, stirring constantly, until fragrant, about 1 minute. Add onion; cook, stirring occasionally, until tender and golden, about 8 minutes. Push onion mixture to sides of skillet. Add eggplants and remaining 2 tablespoons oil to center of skillet; cook, undisturbed, until eggplants are lightly browned on 1 side, about 2 minutes.
Add chickpeas with their liquid, tomatoes, cilantro stems, salt, and garam masala to skillet and stir to combine. Reduce heat to medium-low; simmer, stirring occasionally, until eggplants are tender, about 5 minutes.
Serve eggplant-and-tomato masala over rice, garnished with yogurt and cilantro leaves.