Walnuts, like almonds, start off life on the tree as a soft fruit; as the fruit matures, the nut inside hardens inside a shell. Just like almonds, green walnuts are edible, though their window of availability is even shorter than their gelatinous, amandine cousins. Due to the unbelievably cold summer we've had, the normal June season has arrived… in late August.
While you can eat them plain (peel them), the most popular use for them is unpeeled, quartered and mixed with sugar and vodka to make liqueur de noix (or, as the Italians insist on calling it, nocino); find any limoncello recipe and substitute green walnuts instead, then drink in small quantities as a digestif after a big dinner. Just bear in mind that they will stain your hands.
The season is short; they probably will be available this week, but probably not next week due to the heat.