Melt the butter in a small saucepan over medium-high heat. Add the brown sugar and cook, stirring constantly, until it begins to melt, 2 to 3 minutes. Add the garlic and ginger and cook, continuing to stir, for another 1 to 2 minutes. Stir in the vinegar, fish sauce and soy sauce. Bring to a low simmer and let cook for about 5 minutes. Remove from the heat.
Use a paper towel to thoroughly pat dry the chicken. Sprinkle both sides with salt and pepper.
Heat a large ovenproof skillet over high heat until very hot. Add the oil. Place the chicken in the skillet skin-side down and let brown without moving it for about 4 minutes. Remove the chicken to a plate and raise the heat to medium-high. Add the brown sugar mixture and use a wooden spoon to scrape up any browned bits. Return the chicken to the skillet skin-side up. Transfer the skillet to the oven and bake, basting the chicken with the sauce midway through cooking, until cooked through, about 20 minutes. For extra-crispy skin, transfer the skillet to the broiler for about 2 minutes. Serve over white rice with sauce.