This paper presents AeroTraj, a system that enables fast, accurate, and automated reconstruction of 3D models of large buildings using a drone-mounted LiDAR. LiDAR point clouds can be used directly to assemble 3D models if their positions are accurately determined. AeroTraj uses SLAM for this, but must ensure complete and accurate reconstruction while minimizing drone battery usage. Doing this requires balancing competing constraints: drone speed, height, and orientation. AeroTraj exploits building geometry in designing an optimal trajectory that incorporates these constraints. Even with an optimal trajectory, SLAM's position error can drift over time, so AeroTraj tracks drift in-flight by offloading computations to the cloud and invokes a re-calibration procedure to minimize error. AeroTraj can reconstruct large structures with centimeter-level accuracy and with an average end-to-end latency below 250 ms, significantly outperforming the state of the art.