Heat 2 teaspoons of the oil in a large skillet over medium heat. Season the chicken with ¼ teaspoon each salt and pepper.
Cook the chicken until it is golden brown and cooked through, 7 to 8 minutes per side. Transfer to plates.
Meanwhile, heat the remaining tablespoon of oil in a second skillet over medium-high heat.
Add the apricots, pine nuts, and garlic and cook, stirring, until the pine nuts and garlic are golden brown, 2 to 3 minutes. Remove with a slotted spoon.
Add the broccoli rabe, ¼ cup water, and ½ teaspoon salt to the second skillet and cook, covered, until the broccoli rabe is tender, 2 to 3 minutes.
Add the apricots, pine nuts, and garlic. Serve with the chicken.