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1
Fill a large saucepan three-fourths full with water and bring to a boil.
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2
Add the chicken breasts and return to a boil, skimming off any scum that forms on the surface, then immediately reduce the heat to low.
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3
Simmer, uncovered, until tender, 20 to 25 minutes.
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4
Drain and let cool.
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5
Remove the skin from the chicken breasts, bone the breasts and hand shred the meat with the grain into strips about 1/2 inch thick and 2 inches long.
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6
Refill the large saucepan three-fourths full with water.
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7
Bring to a boil over high heat and add the salt.
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8
Gently pull the strands of noodles apart, then drop them into the boiling water, stirring to separate the strands.
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9
When the water comes to a second boil, boil for 1 minute longer.
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10
Pour the noodles into a colander and rinse thoroughly with cold running water.
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11
Drain thoroughly and transfer to a large bowl.
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12
Toss with the peanut oil to keep the noodles from sticking together.
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13
Set aside.
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14
If using white sesame seeds, toast them in a small, dry frying pan over medium heat until golden and fragrant, about 3 minutes.
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15
If using black sesame seeds, leave them untoasted.
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16
In a large bowl, toss together the chicken, bell pepper, carrot, cilantro and toasted or raw sesame seeds.
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17
Arrange the cooked noodles in a wide shallow bowl.
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18
Scatter the cucumber over the noodles and top with the chicken mixture.
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19
Cover and refrigerate until ready to serve.
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20
To make the dressing, in a small bowl, stir together the peanut butter, sesame paste, sugar, soy sauce and vinegar. In a small saucepan over medium heat, combine the peanut or corn oil, sesame oil and chili oil.
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21
When hot, add the garlic, ginger and green onion; saute gently until fragrant but not browned, about 15 seconds.
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22
Stir in the peanut butter-sesame paste mixture and cook until the mixture begins to form a light syrup, about 1 minute.
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23
Remove from the heat and let cool to lukewarm.
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24
Pour the warm dressing over the chicken salad and sprinkle with the peanuts.
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25
Serve immediately.
