It depends...how much soil / soft material do you have on top of the wood? If it's relatively thin, I expect you'd have more difficulty. If the soil / dirt on top is deep enough, you should be fine. There are other plants that will have deeper and / or broader root systems than many root crops which could actually help tie the hugel together.
There are also variety selections and growing styles you could consider. For instance, 'Paris market' carrots are relatively short and round and should do better in less soil than an imperator type that is long and thin. Cylindra beets are long, but much of them grows above ground, so they may also be OK. I'd imagine radishes should be fine. Potatoes could be grown - if you're concerned about digging them out later, perhaps the
Ruth Stout method with setting the seed potato on the ground, then covering with thick straw or hay could work.
We made raised beds in a hugelkultur style and the carrots and radishes did fine (I don't recall whether we tried beets in the early days). With it being soft and fluffy on top, some of the radishes had over a foot of total root that pulled out. We probably had a good foot of topsoil /
compost at the top as a starting point.
If you do give it a shot, one thing to notice at harvest is whether your carrot roots forked - if so, they encountered something hard and went around it. In ground, that is often a stone, but could be a root or chunk of wood or something similar. That said, roots are powerful. If they have a crack to get into, they can work themselves into your wood in the hugel and can hasten the breakdown of that chunk. Of
course, that could make harvesting a root more challenging.
We helped a bit with the
Boots GAMCOD harvest at
Wheaton Labs in September. Among other things, we harvested potatoes and sunchokes.
If in doubt, and not particularly adventurous, plant above-ground crops or cover crops for a year or two to help develop the soil before switching over to root crops. That said, I'd keep the
permaculture principle about multiple yields in mind, so something you can use, pollinator support, or beauty could be valid yields while helping to build the soil.
Good luck.