Pat chicken dry. Season with salt and pepper.
Heat oil in a large skillet over medium-high heat.
Sear chicken in two batches until lightly browned. Transfer to a plate.
Lower heat to medium. Cook onion and bell pepper until softened, 4–5 minutes.
Stir in ginger and garlic. Cook until fragrant, about 30 seconds.
Blend in curry paste and toast 1 minute, stirring constantly.
Pour in coconut milk and broth. Whisk until the sauce is smooth.
Stir in brown sugar and fish sauce. Return chicken and any juices to the pan.
Simmer gently until chicken is cooked and sauce thickens, 8–10 minutes.
Off heat, stir in lime juice and half the cilantro. Adjust salt to taste.
Serve over rice and garnish with remaining cilantro.