  The existence of compact stellar remnants in the mass range $2-5\,M_{\odot}$
has long been debated. This so-called lower mass gap was initially suggested by
the lack of low-mass X-ray binary observations with accretors in this mass
range, but it has recently been called into question following newer
observations, including a lower-mass-gap candidate with a millisecond pulsar
companion in the dense globular cluster NGC~1851. Here we model NGC~1851 with a
grid of similar dense star clusters utilizing the state-of-the-art Monte Carlo
$N$-body code \texttt{CMC}, and we specifically study the formation of
lower-mass-gap black holes. We demonstrate that both massive star evolution and
dynamical interactions can contribute to forming lower-mass-gap black holes. In
general, the collapse of massive remnants formed through mergers of neutron
stars or massive white dwarfs produces the largest number of lower-mass-gap
black holes among all formation channels. However, in more massive clusters,
supernova core collapse can contribute comparable numbers. Our NGC~1851-like
models can reproduce millisecond pulsar -- lower-mass-gap black hole binaries
similar to the observed system. Additionally, the lower-mass-gap black holes
can also become components of dynamically assembled binaries, and some will be
in merging black hole--neutron star systems similar to the recently detected
gravitational wave source GW230529. However, the corresponding merger rate is
probably $\lesssim 1~{\rm Gpc^{-3}\,yr^{-1}}$.
