Peel mangoes. Cut 1/4-inch-thick slices of flesh away from pits, then cut slices lengthwise into 1/4-inch-thick batons. Gently toss together mangoes, nuoc cham sauce, scallions, and herbs in a large bowl. Let salad stand, stirring occasionally, at least 30 minutes and up to 2 hours. Before serving, stir in peanuts.