1. FOR THE CHICKEN: Pat chicken dry and season with salt and pepper. Heat 2 teaspoons oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add chicken pieces skin-side down and cook without moving until golden brown, 5 to 8 minutes.

2. Using tongs, flip chicken pieces skin-side up. Reduce heat to medium-low, add 3/4 cup broth to skillet, cover, and cook until instant-read thermometer inserted into thickest part of chicken registers 155°F for breasts and 170°F for legs and thighs, 10 to 16 minutes (smaller pieces may cook faster than larger pieces). Transfer chicken to plate, skin-side up.

3. Pour off liquid from skillet into 2-cup measuring cup and reserve. Using tongs, wipe skillet with paper towels. Add remaining teaspoon oil to skillet and heat over medium-high heat until oil is shimmering. Return chicken pieces skin-side down and cook undisturbed until skin is deep golden brown and crisp and it reaches 160°F for breasts and 175°F for legs and thighs, 4 to 7 minutes. Transfer to serving platter and tent loosely with foil. Using spoon, skim fat from reserved cooking liquid and add enough broth to measure 3/4 cup.

4. FOR THE SAUCE: Add 2 tablespoons lime juice, sugar, garlic, fish sauce, and curry paste to now-empty skillet. Cook over medium-high heat, scraping skillet bottom with wooden spoon to loosen browned bits, until spoon leaves wide trail when dragged through sauce, about 2 minutes. Add reserved cooking liquid; return to simmer and cook until reduced to 1/2 cup, 2 to 3 minutes. Stir in any accumulated juices from resting chicken; return to simmer and cook 30 seconds. Off heat, stir in cilantro and remaining tablespoon lime juice. Pour sauce around chicken and serve immediately.

Serves 4