  A precise understanding of the equation of state (EOS) of dense and hot
matter is key to modeling relativistic astrophysical environments, including
core-collapse supernovae (CCSNe), protoneutron star (PNSs) evolution, and
compact binary mergers. In this paper, we extend the microscopic
zero-temperature BL (Bombaci and Logoteta) %nuclear equation of state nuclear
EOS %derived by Bombaci and Logoteta to finite temperature and arbitrary
nuclear composition. We employ this new EOS to describe hot $\beta$-stable
nuclear matter and to compute various structural properties of nonrotating PNS.
%protoneutron stars. We also apply the EOS to perform dynamical simulations of
a spherically symmetric CCSN. The EOS is derived using the finite temperature
extension of the Brueckner--Bethe--Goldstone quantum many-body theory in the
Brueckner--Hartree--Fock approximation. Neutron star properties are computed by
solving the Tolman--Oppenheimer--Volkoff structure equations numerically. The
sperically symmetric CCSN simulations are performed using the AGILE-IDSA code.
Our EOS models are able to reproduce typical features of both PNS and
spherically symmetric CCSN simulations. In addition, our EOS model is
consistent with present measured neutron star masses and particularly with the
masses: $M = 2.01 \pm 0.04 \, M_{\odot}$ and $M = 2.14^{+0.20}_{-0.18} \,
M_{\odot}$ of the neutron stars in PSR~J0348+0432 and PSR J0740+6620
respectively. Finally, we suggest a feasible mechanism to produce low-mass
black holes ($M \sim 2M_{\odot}$) that could have far-reaching consequences for
interpreting the gravitational wave event GW190814 as a BH--BH merger.
