Preheat stove to 450 degrees with rack in upper third. Disperse chickpeas and peppers in the lower part of a shallow, broilerproof 9-by-13-inch baking dish. Mix together spread, hot sauce, paprika, and pepper salt water. Season with salt. Put chicken on top of chickpea blend and brush the two sides with all of spread combination; season with salt and pepper. Cook, skin-side up, until chicken starts to brown and a thermometer embedded into thickest parts (not contacting bones) peruses 160 degrees, 25 to 30 minutes.
Change stove to sear. Brush chicken with container drippings and sear until skin is burned in spots, 2 to 3 minutes. Daintily dress watercress with pressed lemon and skillet drippings. Serve chicken-chickpea blend with watercress, flatbread, and lemon.