Marinate the shrimp: In the bowl of a food processor, combine the ingredients for the marinade, except the shrimp and lime juice, and pulse to form a coarse paste.
In a plastic zipper bag, combine the shrimp and marinade. Seal the bag and gently massage the contents, until the shrimp is thoroughly coated with the paste.
Let marinate in the refrigerator for at least 30 minutes, up to 2 hours. Set the lime juice aside for use later.
For the mango salad: In a small bowl, combine the fish sauce, brown sugar, and lime juice. Stir until the sugar is dissolved.
Put the mango, fresh mint, and chile in a mixing bowl. Add the fish sauce mixture and gently toss to combine. Cover and let chill in the refrigerator.
Start the grill, the broiler, or heat a grill pan on medium-high heat.
Sprinkle shrimp lightly with salt and black pepper.
Place shrimp on the grill rack coated with cooking spray. Grill 2 to 3 minutes per side, turning once, or until shrimp are just cooked through. As you cook, baste the shrimp frequently with the reserved lime juice.
Serve immediately with the chilled mango salad.