Preheat the oven to 350 degrees F (175 degrees C). Place peanuts on a baking sheet.
Toast in the preheated oven until fragrant and browned, 7 to 8 minutes. Let cool for 5 minutes.
Grind roasted peanuts, Thai chilies, garlic, dried shrimp, and sugar in a mortar and pestle or food processor. Stir in lime juice and fish sauce to make dressing.
Toss papaya, tomatoes, green beans, and carrot with dressing. Season with salt and pepper.