  The large-scale distribution of baryons is sensitive to gravitational
collapse, mergers, and galactic feedback. Known as the Cosmic Web, its
large-scale structure (LSS) can be classified as halos, filaments, and voids.
Fast Radio Bursts (FRBs) are extragalactic sources that undergo dispersion
along their propagation paths. They provide insight into ionised matter along
their sightlines via their dispersion measures (DMs), and have been
investigated as probes of the LSS baryon fraction, the diffuse baryon
distribution, and of cosmological parameters.
  We use the cosmological simulation IllustrisTNG to study FRB DMs accumulated
while traversing different types of LSS.
  We combine methods for deriving electron density, classifying LSS, and
tracing FRB sightlines. We identify halos, filaments, voids, and collapsed
structures along random sightlines and calculate their DM contributions.
  We analyse the redshift-evolving cosmological DM components of the Cosmic
Web. We find that the filamentary contribution dominates, increasing from ~71%
to ~80% on average for FRBs originating at z=0.1 vs z=5, while the halo
contribution falls, and the void contribution remains consistent to within ~1%.
The majority of DM variance originates from halos and filaments, potentially
making void-only sightlines more precise probes of cosmological parameters. We
find that, on average, an FRB originating at z=1 will intersect ~1.8 foreground
collapsed structures, increasing to ~12.4 structures for a z=5 FRB. The impact
parameters between our sightlines and TNG structures of any mass appear
consistent with those reported for likely galaxy-intersecting FRBs. However, we
measure lower average accumulated DMs from these structures than the
$\sim90\;{\rm pc\;cm^{-3}}$ DM excesses reported for these literature FRBs,
indicating some DM may arise beyond the structures themselves.
