Many Asian supermarkets have shredded green papaya and that is what I use. However, if you can only find whole green papaya, the papaya can be peeled and shreded using a regular cheese grater with medium to large sized holes. When you get closer to the center, you will see the white immature seeds inside. Stop and move onto another part of the papaya. Discard any seeds that got into your bowl.
In Thailand, green papaya salad is made using a clay mortar, wooden pestle and a spatula. Smash a clove of garlic first. Then add green beans and halved cherry tomatoes. Pound a few times just to bruise the beans and get the juice out of the tomatoes. Add chili peppers and crush them just enough to release the hotness, unless you like your salad really hot. Add the green papaya, dried shrimp, toasted peanuts, fish sauce, lime juice and palm sugar. Use the pestle to push the mixture up in the mortar and the spatula to push it down so that the mixture is mixed well.
However, if you do not have a big enough mortar you can crush garlic, tomatoes, green beans. Set them aside in a large bowl. Add dried shrimp, fish sauce, lime juice and palm sugar to the bowl. Add green papaya and mix well. Serve with sticky rice and a sliver of cabbage, green beans and Thai basil.