As power electronic modules are increasingly required to provide improved heat dissipation, aluminum nitride (AlN) stands out against other ceramic materials. At the same time, more cost-efficient production of customized products demands shorter development cycles and innovative manufacturing processes. Conventional process chains in power electronics are usually long and inflexible; thus, innovative ways to reduce process steps and faster prototyping are needed. Therefore, this study investigates the usage of additive manufacturing technology—laser-based powder bed fusion of metal powder (PBF-LB/M)—namely copper (Cu), on AlN substrates for power electronic applications. It is found that specific electrical conductivity values can be achieved up to 31 MS/m, and adhesion measured by shear testing reaches 15 MPa. In reliability testing, the newly produced samples exhibit a 25% decrease in adhesion after 250 cycles, which is comparatively moderate. This study shows the feasibility of PBF-LB/M of Cu powder on AlN, emphasizing its strengths and highlighting remaining weaknesses.