In a large Dutch oven over medium, heat the oil until shimmering. Add the garlic, chilies and turmeric, then cook, stirring, until fragrant, about 30 seconds. Add the lemon grass, broth, soy sauce, sugar and 1 cup water, then bring to a simmer.
Add the chicken skinned side down in even layer and return to a simmer. Cover, reduce to medium-low and cook until a skewer inserted into the chicken meets no resistance, 30 to 40 minutes.
Using tongs, transfer the chicken skinned side up to a serving bowl. Cook the braising liquid over medium until reduced by about half, about 12 minutes. Remove and discard the lemon grass.
In a small bowl, stir together the cornstarch and 1 tablespoon water. Whisk the mixture into the braising liquid, return to a simmer and cook, stirring constantly, until lightly thickened, about 1 minute.
Off heat, stir the lime juice and fish sauce into the braising liquid, then taste and season with pepper.
Return the chicken and any accumulated juices to the pot, cover and let stand until heated through, about 5 minutes. Return the braise to the serving bowl and sprinkle with cilantro.
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