Make the vinaigrette:
In a small bowl, whisk together the vinaigrette ingredients, seasoning to taste with the salt and pepper.
Cook the chicken:
Heat the peanut oil in an ovenproof 10-inch skillet over medium-high heat. Generously season the chicken breasts with salt and pepper. When the oil is shimmering, add the chicken breasts skin side down and cook until light golden, 1 to 2 minutes. Turn the chicken breasts so they are skin side up and transfer the pan to the oven. Bake until the chicken is just cooked through, 12 to 14 minutes. Transfer the chicken to a cutting board and let cool. Remove and discard the skin and bones. Cut each breast diagonally into thin slices and set aside.
Make the salad:
While the chicken cools, scatter the cashews on a baking sheet and bake in the oven until golden and fragrant, 7 to 9 minutes. Let cool.
Pour the peanut oil into a small saucepan and heat over medium-high heat until the oil reaches 375ºF. Use a candy thermometer to monitor the temperature, or test the oil temperature by dipping the end of one wonton strip into the oil. If it’s hot enough, it should sizzle on contact.
Fry half of the strips until light golden, 30 to 60 seconds. Remove with a slotted spoon and drain on paper towels. Season with a light sprinkle of salt. Fry the remaining wonton strips, drain, and season with salt. (The wontons can be fried up to 2 hours ahead.)
In a large bowl, toss the vinaigrette with the spinach, chicken, and scallions and season to taste with salt. Divide the salad among four or six plates, garnish each with the fried wontons and cashews, and serve immediately.