Controlling the morphology of graphene and other two-dimensional (2D) materials in chemical vapor deposition (CVD) growth is crucial because the morphology reflects the crystal quality of as-synthesized nanomaterials in a certain way, and consequently it indirectly represents the physical properties of 2D materials such as band gap, selective ion transportation, and impermeability. However, precise control of the morphology is limited by the complex formation mechanism and sensitive growth-environment factors of graphene. Therefore, the CVD synthesis of single-crystal hexagonal-shaped graphene islands with specific sizes is challenging. Herein, an unconventional nonlinear carbon supply growth strategy is proposed to realize controllable CVD growth of desired hexagonal graphene islands with specific sizes on Cu substrates. Large-area graphene films of isolated islands with desired densities, sizes, and distances between the islands were successfully synthesized. Subsequently, the direct growth of a planar-tunnel-junction structure based on two parallel gapped graphene islands was achieved by specific adjustment of the growth and etching processes of graphene CVD synthesis. We therefore demonstrated that the nonlinear carbon supply growth strategy is a reliable method for the synthesis of high-quality graphene and can facilitate the direct growth of graphene-based nanodevices in the future.