The manipulation of garments poses research challenges due to their deformable nature and the extensive variability in shapes and sizes. Despite numerous attempts by researchers to address these via approaches involving robot perception and control, there has been a relatively limited interest in resolving it through the co-development of robot hardware. Consequently, the majority of studies employ off-the-shelf grippers in conjunction with dual robot arms to enable bimanual manipulation and high dexterity. However, this dual-arm system increases the overall cost of the robotic system as well as its control complexity in order to tackle robot collisions and other robot coordination issues. As an alternative approach, we propose to enable bimanual cloth manipulation using a single robot arm via novel end effector design -- sharing dexterity skills between manipulator and gripper rather than relying entirely on robot arm coordination. To this end, we introduce a new gripper, called G.O.G., based on a gripper-on-gripper structure where the first gripper independently regulates the span, up to 500mm, between its fingers which are in turn also grippers. These finger grippers consist of a variable friction module that enables two grasping modes: firm and sliding grasps. Household item and cloth object benchmarks are employed to evaluate the performance of the proposed design, encompassing both experiments on the gripper design itself and on cloth manipulation. Experimental results demonstrate the potential of the introduced ideas to undertake a range of bimanual cloth manipulation tasks with a single robot arm. Supplementary material is available at https://sites.google.com/view/gripperongripper.