Instead of dicing cooked chicken breast for a salad, I like to brine a whole chicken, roast it, then pick the meat off the bones—tasty skin bits and all. You end up with tons more flavor. To add another level of yum, I fold in green goddess dressing, an old-school ranch-like classic that’s freshened up with scallions and tarragon.
In a very large pot over medium heat, combine the onion, salt, sugar, garlic, peppercorns, coriander, red pepper flakes, and 4 cups water. Bring the mixture to a boil and whisk until the sugar and salt are fully dissolved. Add the orange and thyme.
Remove from the heat and let cool for 15 minutes. Add the ice cubes to the brine, allow to cool to room temperature, then submerge the chickens. Refrigerate for 8 hours.
Preheat the oven to 425°F. Transfer the chickens to a roasting pan. Discard the brine. Roast the chickens for 30 minutes. Reduce the oven temperature to 375°F and continue roasting for 1 hour more. The chickens’ juices will run clear when the meat is fully cooked. Remove the chickens from the oven and allow them to rest for 20 minutes.
Remove the skin and chop into pieces. Shred the meat into medium pieces. Refrigerate until ready to use.
In a food processor or blender, combine the egg yolks, lemon juice, lime juice, and mustard. Blend until fully combined. While the food processor is running, slowly drizzle in the oil and continue mixing until the dressing thickens. Add the sour cream, scallions, and chopped tarragon. Blend until well combined.
Heat the 1 tablespoon oil in a medium sauté pan over medium-high heat. Cook the onions until tender, about 5 minutes. Transfer to a large bowl and let cool. Add the chicken, chopped skin, dressing, and the basil and tarragon leaves to the onions. Season with salt to taste and toss to coat. (If you plan on doing the preparation ahead, the chicken can be stored in the refrigerator, undressed, for up to 4 days. Dress the salad right before serving.)