Season the chicken with salt and pepper. Roast in a 375-degree oven until an instant-read thermometer inserted in the thigh registers 165 to 170 degrees, about 1 hour, 10 minutes. When cool, pull the meat off the bones in 2-inch shreds, discarding fat and sinew. Reserve 5 cups for this recipe; set the rest aside for another use. (The chicken can be roasted, shredded, and refrigerated in an airtight container for up to 24 hours.).
Puree oil, peanut butter, lime juice, water, 1/4 teaspoon salt, garlic, ginger, brown sugar, and red pepper flakes in blender until combined. Transfer to large bowl. (Dressing may be made ahead of time, covered, and refrigerated overnight. Whisk to recombine before using.).
Add cucumber, carrot, scallions, and cilantro to vinaigrette; toss to combine. Add chicken and toss gently to combine; let stand at room temperature 15 minutes. Adjust seasoning with salt and sprinkle with peanuts. Serve immediately.