Any shallow-rooted plant should do well there. Greens, for instance, would do fine, and any of the varieties of short carrots. Radishes ought to do well in that depth of soil, too.
You could also build a raised bed, just by piling up good soil. Raised beds don't have to have sides, such as boxes. I always just piled up soil on top of the ground to create a rounded mound about 3 or 4 feet wide and however long I wanted it. It's almost imperative to plant that way where I live, because the soil is heavy clay with terrible drainage. To build a new bed, I usually buy topsoil or loam by the truckload from a landscaping supply company. I can get mulch and other stuff at the same time. I mix the soil with compost and spread it about a foot deep where I want to plant. It will settle, so you can spread it even deeper, if you like. I have a friend who makes his new raised beds about 18" deep the first year he plants them. You could just add about 6 to 12 inches to your bed, and then you could plant just about any veggie there.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams