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1
Place the chicken breast-side up in a high-sided pot that fits the chicken snugly.
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2
Fill the pot with cold water to about 1 inch above the chicken and bring to a boil over medium-high heat.
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3
While waiting for the water to boil, cut the ginger into very thin rounds, peel and quarter 1 of the onions and add them and 1 tablespoon salt to the pot.
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4
Once the water reaches a boil, reduce to a simmer and cook, skimming and discarding any scum that floats to the surface of the poaching liquid, until an instant-read thermometer inserted between the breast and thigh of the chicken reads 165 degrees F, 30 to 35 minutes.
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5
(To take the temperature, use a pair of sturdy tongs to carefully lift the chicken out of the hot poaching liquid and insert the thermometer between the breast and thigh.)
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6
While the chicken is cooking, core the white and purple cabbages, very thinly slice them into ribbonlike strands and place the sliced cabbage in a large bowl; set aside.
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7
Fill a second large bowl a little more than halfway with equal parts ice and water, and use the tongs to add the cooked chicken to it; reserve the pot and poaching liquid.
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8
Leave the chicken in the ice water to cool completely, about 10 minutes.
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9
Meanwhile, peel, halve and very thinly slice the remaining onion, and place it in a small bowl.
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10
Add the vinegar, oil, 1 teaspoon of the sugar, 1/4 teaspoon salt and 1/4 teaspoon black pepper.
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11
Use your hands to gently massage the vinegar mixture into the onions until they begin to wilt slightly.
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12
Let the marinated onions sit at room temperature for at least 15 minutes and up to overnight.
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13
Discard the ice water, but keep the chicken in the bowl.
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14
Remove and discard the skin.
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15
Shred the chicken into large bite-size pieces; set aside.
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16
Add the chicken carcass to the poaching liquid, and return it to a simmer.
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17
Cook for 30 minutes, skimming off any fat and/or scum that floats to the surface.
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18
Strain the poaching liquid over a large bowl; discard the solids.
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19
Let this flavorful broth cool completely, and reserve it for another purpose.
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20
(It can be frozen in an airtight container for up to 3 months.)
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21
While the broth is cooking, make the dressing: Cook the remaining sugar and 1/4 cup water in a small saucepan over medium heat, stirring occasionally, until the sugar has completely dissolved, about 5 minutes.
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22
Remove to a small bowl and let cool slightly.
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23
While the sugar mixture is cooling, finely grate the garlic, halve and juice the limes (about 1/2 cup of juice total) and slice the chiles into thin rounds.
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24
Whisk the fish sauce into the cooled sugar mixture; add the garlic, lime juice and chiles.
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25
To assemble the salad, add the chicken, marinated onions (and any residual liquid) and lime dressing to the cabbage bowl.
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26
Tear the mint and basil leaves into the bowl, and gently toss to combine.
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27
Season with additional fish sauce if needed.
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28
Let the salad sit for 15 minutes so the flavors can marry, then transfer to a large platter.
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29
Place the peanuts and fried shallots in separate piles beside the salad.
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30
Just before serving, gently toss together the salad, peanuts and fried shallots and any liquid that has accumulated at the bottom of the platter (the liquid has a lot of great flavor).
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31
Serve with the fried shrimp chips.
