  In this paper we present an alternative cosmological extension of the
three-dimensional extended Newtonian Chern-Simons gravity by switching on the
torsion. The theory is obtained as a non-relativistic limit of an enhancement
and $U(1)$-enlargement of the so-called teleparallel algebra and can be seen as
the teleparallel analogue of the Newtonian gravity theory. The
infinite-dimensional extension of our result is also explored through the Lie
algebra expansion method. An infinite-dimensional torsional Galilean gravity
model is presented which in the vanishing cosmological constant limit
reproduces the infinite-dimensional extension of the Galilean gravity theory.
