When I used to live in NYC, one of my special occasion places was a lovely, pretty serene vegetarian Korean place (HanGawi on 32nd St.). It used a lot of ‘mountain roots’ in the food and while some of the dishes were slightly bland, the overall experience was exceedingly relaxing and special.
So, where in this city of angels, with its huge Korean population, do I find a similar experience? Or even just vegetarian Korean food without the serene atmosphere?
I think Hangawi is one-of-a-kind in the US (afaik, it's the only vegetarian Korean place in the whole country, well along with Franchia, which is the same people). I love Hangawi and would love to see one in LA, but I doubt it will happen any time soon.
I'll occasionally eat at Tofu-Ya with my gf (see above; I would make sure you specify you want a vegetarian broth, btw), but I think in general, Korean food is not so good if you're vegetarian, at least unless you are going with someone who can communicate your food needs to the staff effectively.
On the Westside, on Sawtelle there's Tofu-ya, a soon tofu place. It's not as good as the ones in K-town, but if you don't want to make the drive, it's a decent replacement. They have a vegetable-based soon tofu and some of the panchan is veggie friendly such as the japchae, the beansprouts salad, and the cucumber kimchi. Don't eat the kimchi though, because they make it with shrimp/anchovies, and obviously, not the cold fish cake panchan either.
Strangely, although vegetarian restaurants are not uncommon in Korea (either Buddhist fare or "mountain vegetable" places in the mountains), there aren't any here. Ma Dang Gook Soo serves "san che" (mountain vegetable) bibimbap, though I've never tried it. Oh, and kong gooksu, which they also serve, is vegetarian -- noodles in a cold soy broth. And I think the chic Sohoju in Little Tokyo has something vegetarian... mushroom soontubu or some such, my veggie coworker said he liked it. Often in K restaurants even if a dish features vegetables it still uses meat broth, but I think Sohoju's version is genuinely vegetarian.