Step 1Heat a grill pan or grill. Place the shrimp in a bowl. Add the olive oil, salt and pepper and toss to coat. Grill until opaque in the center, about 3 minutes per side. Remove from the grill, cool and slice in half lengthwise.
Step 2In a small bowl, combine the lime juice, garlic, sugar, fish sauce and chopped chile. Whisk until the sugar is dissolved.
Step 3In a small skillet or saucepan, heat the oil. Fry the minced shallots until golden brown. Drain and add to the fish sauce mixture.
Step 4Julienne the papaya or mangoes into thin, matchstick strips 2 inches long until you have 3 cups. Place in a large serving bowl or platter. Pour the dressing all over the strips, evenly coating them. Toss with 2 tablespoons coriander and 2 tablespoons Thai basil.
Step 5Top the dressed papaya with cooked shrimp and garnish with the peanuts, the remaining 2 tablespoons coriander and the remaining 2 tablespoons basil.