1. To a large bowl, add the fish sauce, soy sauce and cornstarch and whisk until smooth. Add the garlic and chicken, toss to coat and set aside.
2. To a large nonstick skillet set over medium-high heat, add the oil. Once it starts to shimmer, after about 1 minute, add the red onion and chiles. Cook until the red onions brown around the edges, about 2 minutes. Stir in the bok choy stems, cook for 1 minute, then add the chicken. Cook, stirring often, until the chicken is nearly cooked through, 5 to 6 minutes.
3. Stir in the bok choy greens, 1¼ cups of the basil leaves and the lime juice. Once the greens are wilted, transfer the stir-fry to a platter. Sprinkle with the remaining basil leaves and serve.