Aims. We aim to search for transiting exoplanets around the star β Pictoris using high-resolution spectroscopy and Doppler imaging that removes the need for standard star observations. These data were obtained on the VLT with UVES during the course of an observing campaign throughout 2017 that monitored the Hill sphere transit of the exoplanet β Pictoris b. Methods. We utilized line profile tomography as a method for the discovery of transiting exoplanets. By measuring the exoplanet distortion of the stellar line profile, we removed the need for reference star measurements. We demonstrated the method with white noise simulations, and then looked at the case of β Pictoris, which is a δ Scuti pulsator. We describe a method to remove the stellar pulsations and perform a search for any transiting exoplanets in the resultant data set. We injected fake planet transits with varying orbital periods and planet radii into the spectra and determined the recovery fraction. Results. In the photon noise limited case we can recover planets down to a Neptune radius with an ~80% success rate, using an 8 m telescope with a R ~ 100 000 spectrograph and 20 min of observations per night. The pulsations of β Pictoris limit our sensitivity to Jupiter-sized planets, but a pulsation removal algorithm improves this limit to Saturn-sized planets. We present two planet candidates, but argue that their signals are most likely caused by other phenomena. Conclusions. We have demonstrated a method for searching for transiting exoplanets that (i) does not require ancillary calibration observations, (ii) can work on any star whose rotational broadening can be resolved with a high spectral dispersion spectrograph, and (iii) provides the lowest limits so far on the radii of transiting Jupiter-sized exoplanets around β Pictoris with orbital periods from 15 days to 200 days with >50% coverage.