They are rather bitter though. Then again so is orange peel, coffee or chocolate. Chocolate goes well with almost everything, even chicken or cauliflower, so that's something you could try.
I don't like their texture for a dessert. You could turn them into whipped cream, espuma, ice cream, granita or sorbet. Works for basil. Not sure what you'd pair the eggplant with though.
You could make a liquor with them. Just dice and let them soak in a sterilized jar for 2 weeks, covered with a neutral distillate like vodka, grappa or gin. Zabaglione, tiramisu and "flamber" all become an option that way.
Also have a look at http://blog.foodpairing.com/2011/08/chocolate-eggplant/ and http://blog.foodpairing.com/2015/10/aubergine-burger-with-coffee-tamarind-pea-pesto-mint/. All this guy ever does is talk about weird food combinations simply because their aromatic profiles match. Most of the site is for pay, but the freebies are enough to give you an idea, like the mango-eggplant combo.