Chicken Satay Salad

Name: Chicken Satay Salad
Ingredients: 2 skinless, boneless chicken breasts (about 1 pound); 7 tablespoons (about 1/2 cup) Thai peanut sauce; 3 tablespoons vegetable oil; Kosher salt and freshly ground pepper; 2 tablespoons fresh lime juice; 1 head romaine lettuce, sliced; 1/2 English cucumber or 2 Persian cucumbers, cut into matchsticks; 1 medium carrot, halved lengthwise and thinly sliced; 1 red bell pepper, thinly sliced; 1/4 cup roughly chopped fresh cilantro, plus more for topping; 3 tablespoons chopped roasted salted peanuts