Shade tolerant veg

Hello. I have two raised beds which are under trees and receive very little, if any, direct sunlight. Almost a month ago I dug and weeded them (the weeds had been there for some years) and since then only a couple of bits of bindweed have tried to grow at the edges. Other than that nothing, the plots remain clear. I had thought they maybe good for lettuces but am now beginning to wonder if trying this would be a wasted effort. After reading the article on Balsam I considered this may be a good use of the space but as both plots are close to the shed the bees from the hive next door may make life a bit uncomfortable. Would even Balsam grow in such shade. Does anyone have any suggestions please. 
Thank you.

I agree that the leafy veg are likely to be best, but also bear in mind that the trees will not only cast a shade shadow but also a rain shadow. You will need to keep on top of the watering situation quite closely, not least because raised beds of themselves are usually free-draining.

If you keep them watered, shady areas are also useful for growing crops that dislike heat and bolt easily.  Coriander, radish, oriental greens and salad crops may well do well over summer and early autumn.

pigeons don't like crispy lettuce

Cabbages and flat leaf parsley, coriander, oriental leaves..pak choi, oh and green dwarf beans have done well for me