There are many kinds of vegetables that will re-grow from kitchen scraps
Growing Basil from seeds takes a long time and the seedlings are tender to frost and slugs. One alternative is to grow them from cuttings.
A full grown Basil plant, from the store or previously grown
Kitchen scissors or a sharp knife
A glass or cup
Online there are many suggestions that you will need to use rooting hormone or alike. This is not required at all.
Take a cutting of around 10 cm long, cutting right under the node where the leaves come out of.
Remove the big leaves at the bottom (you can use them in cooking or salads) but leave the top ones and any little side shoots
Place the cuttings in the glass of water where there is sunshine
Watch the water levels carefully, making sure that they do not dry out.
After 5-7 days you will see some roots appearing you can keep them in the water for another week and you can plant them out into some dirt.