How do I roast eggplants without a gas stove?
You have several options:


Use the the broiler (or grill, as they say in the UK).
Use a wok ring or similar to suspend the eggplant above your electric element.  Set the element to its highest setting, cooking the eggplant by infrared.  It just may take longer than the gas element would have.   If you don't have a wok  ring, you can use aluminum foil crushed to improvise an eggplant stand.
Buy a small hibachi and use it&mdash;the original recipe is designed for a grill (in the US sense, barbecue in the UK parlance).