How to Make It
In a small skillet, toast the rice over moderate heat, tossing, until lightly browned, about 4 minutes. Transfer the rice to a mortar or spice grinder and let cool completely. Pound or grind to a powder.
In a large mortar (or food processor), pound the dried shrimp until coarsely ground. Add the garlic, chile and peanuts and pound coarsely. Add both sugars and pound to a paste. Add the tomatoes and pound to coarsely crush them. Stir in the lime juice and fish sauce.
In a large bowl, toss the green papaya, carrot, shallots, cilantro and mint leaves. Add the tomato dressing and toss well.
In a saucepan of boiling water, add the drained rice noodles. Cook until just al dente, about 30 seconds. Drain and return the noodles to the saucepan. Fill the saucepan with cold water and swirl. Drain and repeat 3 times to prevent the noodles from sticking together. Drain the noodles in a colander and lift them up a few times to release excess water.
Heat a medium skillet. Season the duck breasts with salt and add them to the hot skillet, skin side down. Cook over moderately high heat for 2 minutes. Reduce the heat to moderate and cook until the skin is browned and crisp, about 2 minutes longer. Turn the duck and cook until medium-rare, about 4 minutes longer. Transfer the duck to a carving board to rest for 5 minutes. Thinly slice the duck breasts crosswise.
Add the rice noodles to the green papaya salad and toss well. Transfer to a platter. Arrange the duck slices on top. Sprinkle with the toasted rice powder and serve.
Green (unripe) papayas have dark green skin and are very firm to the touch.