Prototyping compact devices with unique form factors often requires the PCB manufacturing process to be outsourced, which can be expensive and time-consuming. In this paper, we present Fibercuit, a set of rapid prototyping techniques to fabricate high-resolution, flexible circuits on-demand using a fiber laser engraver. We showcase techniques that can laser cut copper-based composites to form fine-pitch conductive traces, laser fold copper substrates that can form kirigami structures, and laser solder surface-mount electrical components using off-the-shelf soldering pastes. Combined with our software pipeline, an end user can design and fabricate flexible circuits which are dual-layer and three-dimensional, thereby exhibiting a wide range of form factors. We demonstrate Fibercuit by showcasing a set of examples, including a custom dice, flex cables, custom end-stop switches, electromagnetic coils, LED earrings and a circuit in the form of kirigami crane.