Using a small, sharp knife or scissors, slit each prawn open lengthwise down the back from top to tail, taking care not to cut all the way through. Remove the veins. Press each prawn gently to flatten open. Place the prawns on a rimmed baking sheet with shell sides down.
To make the pesto, place the oil, garlic, ginger, lime juice, chili paste, and salt in a food processor. Process until smooth. Remove 1/4 cup of the oil mixture and brush onto the prawns.
Roast the prawns on the top rack of an oven preheated to 400°F. Cook 5 to 7 minutes, until the prawns are pink and opaque.
While the prawns are roasting, add the cilantro, peanuts, and toasted sesame oil to the food processor with the remaining oil mixture. Process until a pastelike consistency is formed.