Place chicken breasts between two sheets of parchment or plastic wrap. Using a mallet or rolling pin, pound each to an even thickness of 1/2". Do not make any thinner or they could dry out.
In a large bowl, whisk together soy sauce, fish sauce, ginger and garlic. Grate the zest of 1 lime and squeeze in its juice. Whisk in peanut oil. Place chicken breasts in bowl and turn breasts well to evenly coat with mixture. Cover and refrigerate for at least 1 hour and up to 4 hours. remove chicken from fridge while you heat the grill.
Light the grill, building a hot fire, or heat your gas grill to high. Once the grill is fully heated, brush breasts lightly with peanut oil and place chicken on the grill. Cook until undersides are browned and chicken is about halve way cooked. Flip breasts and grilled until cooked though, 3 to 5 minutes more.
Transfer chicken to a platter. Drizzle with sesame oil, garnish with lime juice and cilantro and chiles if desired.