I have tried regrowing a few times but haven't been that successful. Look for strong fresh organic plants such as those in home delivered fruit and vegetable boxes. Better still harvest your own so that you can leave a bit more of the structure you need for the regrowth.
Some plant parts you can stick in water or soil to reroot and an entire new plant will grow e.g. garlic. With some plants the plant will continue to grow from a leftover piece e.g. lettuce. Some plants will grow something different such as carrot tops and beetroot leaves.
If the plant requires soil it is better to use soilless mix which is sterile. It should be damp but not wet.
There are also some ideas for seeds to use from organic produce here
Here are some plants to try:
Celery, Bok Choy, Cabbage and Romain Lettuce
Look for plants that have the bottom structure in place cut off the leaves and but leave a couple of inches so that the growing tip is present. Sit the base in a shallow container with water. You must refresh this water regularly to keep the plant healthy. You can suspend the head on cocktail sticks above the water line. The top of the plant should begin to regrow. Once regrowing you can snip of leaves for salad or plant in good quality organic soil and water regularly. Harvest once the plant is mature.
Beetroot, Carrot, Radish, Turnip
These root vegetables will regrow their leaves. This can be done in a saucer of water but the water will need to be replaced daily and even then the pieces can still get slimy.
Carrot - cut off about 2 cms of carrot cut the leaves to less than a centimetre. Plant in soil but do not cover the tops. Keep the soil moist and snip of the leaves when you need them. You might even get a flower. Carrot tops can be used like parsley or made into a pesto.
The same process can be used for the other vegetables. Beetroot and radish leaves can be used in salads but turnip leaves are better lightly cooked.
Lemongrass, Leeks, Fennel
Use a taller container or vase. Change the water daily and watch for the root regrowth. Once established plant in organic soil and use the greenery when needed.
Garlic regrows from a single clove but will take nearly a year.
Ginger and Turmeric
Plant a piece in well drained organic soil with the eyes level with the surface and leave in a sunny spot indoors or in the garden if the temperature is consistently above 20 degrees. Green tips should appear in a few weeks. Once established ginger can be grown outside in a sheltered area it is an understory plant which likes dappled light in a hot humid environment. It takes about a year and will flower. To harvest cut a little from the root disturbing the plant as little as you can and recover with soil. It grows very well in containers and this will give you the ability to move the plant around the garden to suit the conditions.
If a potato has sprouted you can plant the whole potato or cut it so each piece has at least two eyes. Plant in soil.
This process is similar to the way an avocado stone grows. In early Spring suspend the potato over a little water change regularly and do not allow to dry out. Roots will grow and then little stems will grow from the top of the potato these are the slips. When the slips are 3 to 4 inches long remove and put into a jar of water. The slips will form foots once established plant into a pot and plant out once the last frost has been and harvest in October.
Cut the root section leaving a generous amount of leek and plant so a centimetre of stalk is above the soil. The green parts will grow and can be used to flavour salads and soups.
Twist off the leaves by holding the at the base. They should come off with a little plug of pineapple. Trim the bottom third of the leaves off to expose the stem. Cut about half an inch off and remove any remaining bits of fruit. Suspend in water until new roots and a new leaf has grown. It can now be planted into a pot. It needs to be kept in bright light indoors. After two years you might get a fruit!
Plants that I have been successful with:
When onions sprout you can carefully unwrap the onion to find little bulblet in the center, these can be planted with the roots in the soil an inch deep and the growing tip above the soil. You can grow on to form a larger bulb or use the onion leaves to add flavour to dishes.
Shallots are slightly different they have cloves inside similar to garlic and can be grown on in the same way.