Laser machining fired ceramic and other hard and/or thick materials includes scribing a workpiece with a laser beam along a sequence of parallel laser paths within a cutout region of the workpiece. The scribing creates a kerf in the cutout region that widens as the laser beam advances along the sequence. The sequence may begin with an inner portion of the cutout region and end with an outer edge thereof such that debris is directed away from the laser paths to increase throughput and create a high quality opening in the workpiece. High quality structures may also be cut out from the workpiece. The method includes directing a high velocity stream of gas to an interface of the laser beam and the workpiece to redirect the flow of debris and cool the interface. The method may also adjust a focus depth of the laser beam as it deepens the kerf.