Next-best would be quick-growing small root crops (like radishes, turnips, and green onions) --- and if they kinda "under-produce" in the edible root dept., you can still eat the greens.
What you don't want is plants grown for fruits or seeds: that would rule out squash, as well as melons, cukes, tomatoes, peppers, corn, sunflowers, pole beans, etc. Anything that makes a fruit or seeds, needs lots and lots of sun.
Peas? I'm thinking no: even if the vines (and leaves) grew OK, the yield would be low, probably a big disappointment.
Nix on potatoes or shallots or onions: they will not tolerate interfering tree-roots. And as far as I know, almost no garden plant will tolerate walnut tree roots: they are notoriously inhospitable to competition.
Bottom line: in shady areas, stick with greens. In my humble opinion.
Thanks for the great information!