In a large bowl, combine the fish sauce, soy sauce, and cornstarch and whisk until smooth. Add the garlic and chicken, toss to coat, and set aside.
Add two teaspoons of oil to a large nonstick skillet set over medium-high heat. Once it starts to shimmer, add the red onion and chiles. Cook until the 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.
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.