Step 1

Put the kaffir lime leaves and 1/3 cup of the oil in a blender. Set at the blender's lowest speed, start the blender and slowly increase the speed to high. Blend for 25 seconds. Transfer to a small bowl and let sit for 5 minutes. Strain through a fine-mesh sieve into a clean small bowl and reserve. Clean the blender carafe.

Step 2

Put the cilantro, chiles, garlic, shallots, ginger, lemongrass, 1/2 teaspoon salt and 1/3 cup water in the blender. Set at the blender's lowest speed, start the blender and slowly increase the speed to medium-high. Blend for 40 seconds to make a fine puree, stopping once to scrape down the sides of the carafe.

Step 3

Heat the remaining 1/4 cup oil in a large high-sided skillet over medium heat. Carefully add the cilantro puree. Cook, stirring constantly, until thick and olive green, 10 to 12 minutes. Add the fish sauce and brown sugar. Cook 1 minute more, then add the coconut milk and 1 1/4 cups water. Add more salt and fish sauce if desired. Bring to a simmer and cook gently for 5 minutes.

Step 4

Turn the heat to medium-low and add the shrimp and lime juice. Cover and cook until the undersides of the shrimp are beginning to turn pink, about 3 minutes. Flip the shrimp, cover and cook for 2 minutes more. Add the tomatoes and snow peas, cover and cook until the shrimp are pink on both sides and cooked through, about 1 minute more.

Step 5

Ladle the curry into 4 soup bowls. Sprinkle with cilantro leaves and drizzle with some of the kaffir lime oil.