  We search for transiting exoplanets around the star $\beta$ 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 $\beta$ Pictoris b. We utilize line
profile tomography as a method for the discovery of transiting exoplanets. By
measuring the exoplanet distortion of the stellar line profile, we remove the
need for reference star measurements. We demonstrate the method with white
noise simulations, and then look at the case of $\beta$ Pictoris, which is a
$\delta$ 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 inject fake planet transits with varying orbital periods and planet radii
into the spectra and determine the recovery fraction. In the photon noise
limited case we can recover planets down to a Neptune radius with an $\sim$80%
success rate, using an 8 m telescope with a $R\sim 100,000$ spectrograph and 20
minutes of observations per night. The pulsations of $\beta$ 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. 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 $\beta$ Pictoris with orbital
periods from 15 days to 200 days with >50% coverage.
