Okay, I admit it. I recycle even plants

You are not alone when it comes to stuff like this. :D
Actually love to see how most of those "left over roots" go wild again. Some look different than normally, but they taste the same. :)

It worked very well with cabbage, too, though the cabbage is growing higher and higher, and not really bigger. That happens to some of the onions, too, and I'm not sure why.

Sadly most of those plants you can get from the supermarket have manipulated genetics, so you will never be able to regrow them properly. Also tried to regrow cabbage when i saw it starting to grow out of a smaller piece that was left over while standing inside the fridge.
Was so fascinated that it was growing inside the fridge, so i gave it a try. But yea, it simply was growing high, had some leaves, flowered again and got seeds, but NONE of those new plants that were growing out of the seeds ever evolved into a cabbage (head). It was just some leaves and flowered again - watched 3 or 4 generations going like this and then decided to feed it to the chickens.^^

Have a nice day and wish you good luck on your vertical garden project - keep us up to date. :)