The Search for Extraterrestrial Intelligence has traditionally been conducted at radio wavelengths, but optical searches are well-motivated and increasingly feasible due to the growing availability of high-resolution spectroscopy. We present a data analysis pipeline to search Automated Planet Finder (APF) spectroscopic observations from the Levy Spectrometer for intense, persistent, narrow-bandwidth optical lasers. We describe the processing of the spectra, the laser search algorithm, and the results of our laser search on 1983 spectra of 388 stars as part of the Breakthrough Listen search for technosignatures. We utilize an empirical spectra-matching algorithm called SpecMatch-Emp to produce residuals between each target spectrum and a set of best-matching catalog spectra, which provides the basis for a more sensitive search than previously possible. We verify that SpecMatch-Emp performs well on APF-Levy spectra by calibrating the stellar properties derived by the algorithm against the SpecMatch-Emp library and against Gaia catalog values. We leverage our unique observing strategy, which produces multiple spectra of each target per night of observing, to increase our detection sensitivity by programmatically rejecting events that do not persist between observations. With our laser search algorithm, we achieve a sensitivity equivalent to the ability to detect an 84 kW laser at the median distance of a star in our data set (78.5 ly). We present the methodology and vetting of our laser search, finding no convincing candidates consistent with potential laser emission in our target sample.