Pat chicken dry and season with salt and pepper. Heat 2 tsp oil in a large skillet over medium high heat until shimmering. Add chicken pieces skin side down and cook without moving until golden brown, 5-8 minutes.
Using tongs, flip chicken pieces skin side up. Reduce heat to medium low, add broth to skillet, cover, and cook until it reaches an internal temperature of 165°, 10-16 minutes. Transfer chicken to a plate.
Pour off liquid into 2-cup measuring cup and reserve. Wipe skillet with paper towels. Add remaining teaspoon oil and heat over medium high heat until it is shimmering. Return chicken skin side down to skillet and cook undisturbed until skin is deep golden brown and crisp and internal temp is 160°, 4-7 minutes. Transfer to heated plates and tent loosely with foil. Skim fat from reserved cooking liquid and add enough additional broth to yield 3/4 cup.
To the skillet add 1/2 the lime juice, sugar, garlic, fish sauce and curry paste. Cook over medium high heat, scraping bottom of the pan with a wooden spoon until the spoon leaves a wide trail in the sauce, about 2 minutes.
Add reserved cooking liquid; return to simmer and cook until reduced to 1/3 cup, 2-3 minutes. Stir in any juices from the resting chicken; return to simmer and cook 30 seconds. Off heat, stir in cilantro and remaining lime juice. Pour sauce around chicken and serve immediately.