Preheat your broiler to its highest setting.
Line a baking sheet with aluminum foil for easy cleanup.
Pierce the eggplants all over with a fork and prick the tomatoes in a few spots.
Place them on the prepared baking sheet and position it as close to the broiler's heat source as possible.
Broil the vegetables, turning them occasionally, until the skins are charred and blistered all over and the flesh is completely soft. Slender eggplants and tomatoes will take about 15–20 minutes.
Remove them from the oven and let them cool in a sieve set over a bowl until they are comfortable to handle.
Once cooled, carefully peel the charred skin off the eggplants and tomatoes. Finely chop the smoky pulp and transfer it to a mixing bowl.
Add the salt, black pepper, green chilies, shallots, mint, cilantro, lime juice, and 2 teaspoons (10 ml / 0.3 fl oz) of the mustard oil to the bowl.
Mix everything thoroughly until well combined.
Taste and adjust the seasonings as needed.
Transfer the chokah to a serving bowl.
Drizzle the remaining 1 teaspoon (5 ml / 0.2 fl oz) of mustard oil over the top before serving.
Enjoy it cold or at room temperature.