Laser brightness is a measure of the ability to deliver intense light to a target and encapsulates both the energy content and the beam quality. High-brightness lasers require that both parameters be maximized, yet standard laser cavities do not allow this. For example, multimode beams, a mix of many transverse modes, have a high energy content but low beam quality, while single transverse mode Gaussian beams have a good beam quality, but their small mode volume means a low energy extraction. Here we overcome this fundamental limitation and demonstrate an optimal approach to realizing high-brightness lasers. We employ intra-cavity beam shaping to produce a single transverse mode that changes profile inside the cavity, Gaussian at the output end and flattop at the gain end, such that both energy extraction and beam quality are simultaneously optimized. This work should have a significant influence on the design of future high-brightness laser cavities.