This is a bit more involved than my favorite recipes tend to be, but it is so delicious.
The two things you might not be able to find at your regular grocery store are vegan margarine and plain soy yogurt; but plenty of grocery stores do have those.
If yours doesn't: I haven't tried this with olive oil instead of margarine, but it would probably be fine. And I'm pretty sure I made a vegan "buttermilk" in place of the soy yogurt. I'd guess it's much better with the yogurt, but my version worked well enough for me to enjoy it.
To make vegan buttermilk or yogurt substitute (usually for baking, as in buttermilk biscuits), just mix a cup of soy milk with a tablespoon of vinegar. I do recommend getting yogurt for this if you can, though.
This is another recipe in which a combination of simple ingredients makes something amazing happen.
Iranian Rice with Beans and Dill (Baqala Polo)