Peel about half of the green papaya. One whole papaya will yield a lot of shredded papaya. I like to peel the portion that I will be using and keeping the rest unpeeled so it stays fresh in the refrigerator, wrapped in plastic wrap.
While being very careful, and possibly using a towel to protect the hand you are holding the papaya with, use a sharp knife to make lots of cuts about 1/2 inch deep all over the peeled section of papaya.
The cuts should all run lengthwise in the same direction and be closely spaced together.
Use the knife to cut away from you and "shave" the papaya so that shredded bits of papaya fall away onto the cutting board. This may take a bit of practice. You want there to be a little variance in the size of the shreds. Don't shave the papaya extremely finely or the texture will not be as pleasant and the salad will be overly soggy.
Keep shredding and turning the papaya until you have shredded the papaya almost down to the seeds in the center. You will probably need to make fresh cuts into the papaya after you have shaved off some of the outer parts of the papaya. You want to yield 2 generous cups of shredded papaya.
Place the dry shrimp into the mortar and pestle and pound them until they start to break apart and get a bit fibrous.
Add the garlic, bird chilies, and sugar to the mortar and pestle and pound into a rough paste. Don't make this paste super fine.
Cut the reserved, already squeezed lime into 4 pieces and put it in a mixing bowl along with the shredded papaya, cut long beans, and halved tomatoes.
Add the contents of the mortar and pestle to the mixing bowl along with 3 Tbsp lime juice, 1.5 Tbsp fish sauce, and 2 Tbsp prepared tamarind. Use the pestle from your mortar and pestle or any other heavy blunt utensil such as a rolling pin and pound the ingredients in the salad while mixing. The tomatoes should be releasing their juices and the papaya should be slightly bruised. Lightly pound the pieces of lime to expel some of the oils from their skins (the pieces of lime will be the only inedible component in the salad).
Add the peanuts to the mixing bowl.
Give the salad a light final pound to just break up some of the peanuts and mix well. Adjust seasoning if necessary with more fish sauce, lime juice, or sugar. Salad should be salty, sour, spicy, and with a very light sweetness... just enough to soften up the other aggressive flavors.
Serve the papaya salad with sticky rice.