Preheat the oven to 350 degrees F. Put the pears on a baking sheet, cut sides up. Sprinkle with salt and pepper and drizzle with 1 tablespoon sesame oil. Place them into the oven and roast 15 to 20 minutes, until very tender when pierced with a knife.
Meanwhile, put the duck breasts on a cutting board skin side up and score all over in a tiny crosshatch pattern, so the fat will render and the skin will crisp. Season all over with salt and pepper and drizzle with 1 tablespoon sesame oil. Add the breasts to a stainless steel pan over medium heat, skin side down, and cook slowly for about 12 to 15 minutes until the fat is rendered and the skin is browned and crispy. Occasionally spoon the fat out into a bowl. Turn the breasts and cook 1 to 2 more minutes for medium-rare. Transfer the duck to a cutting board to rest.
To serve, cut the duck breasts crosswise, on an angle, into 8 slices each. Mound the Sesame Noodles evenly between 4 bowls. Add half a sliced duck breast to each bowl along with a peach half. Garnish with watercress and lime wedges.
Cold Sesame Noodles:
Bring a large pot of salted water to a boil over high heat. Add the noodles and stir to keep them from sticking together. Cook until barely tender and still firm, 3 to 4 minutes. Drain immediately and rinse under cold running water to cool. Drain the noodles really well, transfer to a wide serving bowl, and toss immediately with 3 tablespoons of the sesame oil so they don't stick together.
Meanwhile, make the sauce. In a saucepan, heat 1/4 cup sesame oil over medium-low heat. Add the ginger, garlic, and chile. Cook, stirring for a minute, until the vegetables are soft and fragrant. Place into a blender along with the brown sugar, peanut butter, vinegar, soy sauce, hot water, chili sauce, and the remaining 2 tablespoons sesame oil. Puree and refrigerate until cold.
Toast the sesame seeds in a dry skillet over medium-low heat until golden brown. (Shake the pan constantly to keep the seeds from burning). Put the noodles in a serving bowl and toss with the sesame sauce. Sprinkle with the sesame seeds and green onions, and serve with the lime wedges.