  Polarimetric observations of Fast Radio Bursts (FRBs) are a powerful resource
for better understanding these mysterious sources by directly probing the
emission mechanism of the source and the magneto-ionic properties of its
environment. We present a pipeline for analysing the polarized signal of FRBs
captured by the triggered baseband recording system operating on the FRB survey
of The Canadian Hydrogen Intensity Mapping Experiment (CHIME/FRB). Using a
combination of simulated and real FRB events, we summarize the main features of
the pipeline and highlight the dominant systematics affecting the polarized
signal. We compare parametric (QU-fitting) and non-parametric (rotation measure
synthesis) methods for determining the Faraday rotation measure (RM) and find
the latter method susceptible to systematic errors from known instrumental
effects of CHIME/FRB observations. These errors include a leakage artefact that
appears as polarized signal near $\rm{RM\sim 0 \; rad \, m^{-2}}$ and an RM
sign ambiguity introduced by path length differences in the system's
electronics. We apply the pipeline to a bright burst previously reported by
\citet[FRB 20191219F;][]{Leung2021}, detecting an $\mathrm{RM}$ of $\rm{+6.074
\pm 0.006 \pm 0.050 \; rad \, m^{-2}}$ with a significant linear polarized
fraction ($\gtrsim0.87$) and strong evidence for a non-negligible circularly
polarized component. Finally, we introduce an RM search method that employs a
phase-coherent de-rotation algorithm to correct for intra-channel
depolarization in data that retain electric field phase information, and
successfully apply it to an unpublished FRB, FRB 20200917A, measuring an
$\mathrm{RM}$ of $\rm{-1294.47 \pm 0.10 \pm 0.05 \; rad \, m^{-2}}$ (the second
largest unambiguous RM detection from any FRB source observed to date).
