  Twisted bilayer graphene (TBG) hosts nearly flat bands with narrow bandwidths
of a few meV at certain {\em magic} twist angles. Here we show that in twisted
gapped Dirac material bilayers, or massive twisted bilayer graphenes (MTBG),
isolated nearly flat bands below a threshold bandwidth $W_c$ are expected for
continuous small twist angles up to a critical $\theta_c$ depending on the
flatness of the original bands and the interlayer coupling strength. Narrow
bandwidths of $W \lesssim $30 meV are expected for $\theta \lesssim 3^{\circ} $
for twisted Dirac materials with intrinsic gaps of $\sim 2$ eV that finds
realization in monolayers of gapped transition metal dichalcogenides (TMDC),
silicon carbide (SiC) among others, and even narrower bandwidths in hexagonal
boron nitride (BN) whose gaps are $\sim 5$ eV, while twisted graphene systems
with smaller gaps of a few tens of meV, e.g. due to alignment with hexagonal
boron nitride, show vestiges of the magic angles behavior in the bandwidth
evolution. The phase diagram of finite valley Chern numbers of the isolated
moire bands expands with increasing difference between the sublattice selective
interlayer tunneling parameters. The valley contrasting circular dichroism for
interband optical transitions is constructive near $0^{\circ}$ and destructive
near $60^{\circ}$ alignments and can be tuned through electric field and gate
driven polarization of the mini-valleys. Combining massive Dirac materials with
various intrinsic gaps, Fermi velocities, interlayer tunneling strengths
suggests optimistic prospects of increasing $\theta_c$ and achieving correlated
states with large $U/W$ effective interaction versus bandwidth ratios.
