This study presents an automated shape optimization method for heat sinks. The computational framework has been developed by combining a conjugate heat transfer solver with adjoint capabilities, a CAD parametrization tool, and a gradient-based optimizer. The test case considers the design optimization of a water-cooled heat sink with pin fins, with the goal of concurrently enhancing heat transfer and minimizing pressure losses. Results show that the optimized fin geometry leads to an improvement of the average heat transfer coefficient by 24% while the pressure drop is lowered by 19%. The optimal fin array features an unconventional shape with an enlarged cross-section at the hub and the top with respect to mid-span, and a variation of the pin profile in the streamwise direction. The net effect is a reduction in flow blockage, an increase in fin efficiency, and a lower and more uniform temperature distribution in the heat sink base plate.