If you haven't already, prepare vegan fish sauce and set aside (find link above).
See notes for tips on finding a green papaya (mine wasn't quite green, but it still worked!). Then move onto prepping the vegetables.
Once you've peeled your carrots, cut in half crosswise, and then cut those pieces in half lengthwise. Then use a mandolin* with a fine-tooth blade (or julienne peeler) to carefully shred the carrots into small strips (see photo). Continue until all carrots are shredded. Add to a mixing bowl.
To prepare papaya, peel off the skin using a vegetable peeler, then halve lengthwise and scoop out seeds.
Cut into small segments and shred into small strips using your mandoline or julienne peeler. Add to mixing bowl.
Chop cabbage and tomatoes and add to mixing bowl as well. Set aside.
To prepare sauce, add garlic, peanuts, chilies, coconut sugar, and salt to a mortar or small food processor and mash or pulse until it's a fine paste.
Transfer to a small mixing bowl and add lime juice and vegan fish sauce. Whisk to combine, then sample and adjust seasonings as needed.
Add sauce to vegetables and toss to combine. To serve, lay down romaine lettuce on serving dish(es) (optional), and top with papaya salad. Garnish with additional crushed peanuts and lime wedges.
Best when fresh, though leftovers keep covered in the refrigerator for 2-3 days. Papaya salad would make an excellent side dish to spring rolls and Pad Thai!
*It can be challenging to find an underripe, green papaya unless you go to an Asian or specialty store. So, if you can't find one, simply buy the greenest papaya you can find, and use a vegetable peeler to remove the skin. Then halve it, and scoop out the seeds and most of the ripe flesh (save for other uses). Then chop into segments and finely shred with a mandoline or vegetable julienne peeler. *If you don't have a vegetable julienne peeler or mandoline, you can alternatively use a large sharp knife and chop the papaya and carrot very finely, though this takes quite a bit longer. *2 garlic cloves yield ~1 Tbsp minced garlic. *Recipe adapted from The New York Times. *Nutrition information is a rough estimate.