Bring a large pot of water to a boil. In a large skillet, heat 2 tbsp. olive oil over medium. Season the chicken with salt and pepper. Cook until golden on the bottom, about 4 minutes. Turn; add the oregano, butter and half the garlic. Cook, basting with the butter, until cooked through, about 3 minutes. Transfer to a plate.
In the same skillet, cook the onion over medium, stirring often, until softened, 5 to 6 minutes. Discard the oregano; transfer to a food processor. Add the chickpeas, stock and 1/4 cup oil. Puree and season.
Salt the boiling water. Add the broccoli rabe. Cook 1 minute; drain. In the same skillet, heat the remaining 2 tbsp. oil over medium; add the broccoli rabe, crushed red pepper and remaining garlic. Cook, stirring, about 2 minutes; season. Divide the chickpea mixture among plates; top with broccoli rabe and chicken.