Season chicken breasts with salt and pepper. In 12-inch skillet over medium heat, cook chicken breasts in hot oil until fork tender and browned on both sides (about 10 minutes). Arrange on platter and keep warm.
Spoon off the oil in skillet. Squeeze juice from lime into the skillet. Over low heat, cook until lime juice is simmering. With a wire whisk, mix in the butter- 1 tablespoon at a time; cook, stirring until butter becomes opaque and forms a creamy sauce. Stir in the dill and chopped chives.
Spoon sauce over chicken and serve.