Special equipment: Asian shredder Grill pan
Using a paring knife, cut through the back shell of each shrimp. Remove the dark vein by running your forefinger between the shell and the flesh of each shrimp, keeping the shell intact. Reserve shrimp in a re-sealable gallon plastic bag.
In a medium bowl, whisk together the lime juice and sugar until the sugar is completely dissolved, then add the fish sauce. Add the garlic, chile, Thai basil, and 1 tablespoon oil. Stir well and pour over the reserved shrimp in the plastic bag. Seal the bag, squeezing out the air. Massage the bag to make sure that all of the shrimp are covered and refrigerate for 30 minutes to 1 hour.
Heat a grill pan over high heat. Oil the grill pan with remaining oil. Grill the shrimp, turning once to prevent burning, until evenly pink and golden on both sides, 2 to 3 minutes. Serve shrimp on large plate over Sour Mango Salad.
Mix the mangos, carrots, and chiles in a large bowl.
For the dressing: in a medium bowl, whisk together the lime, fish sauce, and sugar until the sugar is completely dissolved. Add the dressing to the bowl and toss together. Let stand 20 minutes. Drain and serve at room temperature or lightly chilled.
Not all ingredients are created equal, so adjust the seasoning as you go. Taste the dressing prior to adding to the ingredients to make sure the sour, salty, and sweet notes are balanced. Adjust the seasoning according to the acidity and sugar levels of the mangos and the salt content of the fish sauce.