What vegetables grow in mostly shade?

I would like to have a garden in my front yard, but our house is surrounded by shade trees. We are even struggling to grow grass!

I put tomato and peppers in the only sunny spot I could find. But I want to grow more!
Update: Also considering growing something on the fence in my back yard, as long as it is safe for my puppy to eat. I would plant it on the outside of the chain link fence and let it climb up.
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