almost any thing, young walnuts will not throw much shade . even as they grow older as long as they are well spaced they have a very open
canopy so any thing that
should grow in shade should grow for you .and despite the folk-law and chemistry that says things should not or will not grow under walnut trees we are on walnut st with a farm full of walnut trees with grass weeds and flowers growing right up to the trees .
I would start by trying beets , chard,lettuce,green onions , radish ,or hosta ! (which I just found out are edible) for deeper shade .
we have 1 baby walnut ( baby being about 10 years and about 20 feet tall near the veg garden but it is close to the tree line . but on the front
lawn my iris sun chokes and butterfly bushes are right under a clump of taller walnuts.
Sue