  We introduce an unrolled quantization $U_q^E(\mathfrak{gl}(1 \vert 1))$ of
the complex Lie superalgebra $\mathfrak{gl}(1 \vert 1)$ and use its categories
of weight modules to construct and study new three dimensional non-semisimple
topological quantum field theories. These theories are defined on categories of
cobordisms which are decorated by ribbon graphs and cohomology classes and take
values in categories of graded super vector spaces. Computations in these
theories are enabled by a detailed study of the representation theory of
$U_q^E(\mathfrak{gl}(1 \vert 1))$, both for generic and root of unity $q$. We
argue that by restricting to subcategories of integral weight modules we obtain
topological quantum field theories which are mathematical models of
Chern--Simons theories with gauge supergroups $\mathfrak{psl}(1 \vert 1)$ and
$\mathfrak{gl}(1 \vert 1)$ coupled to background flat
$\mathbb{C}^{\times}$-connections, as studied in the physics literature by
Rozansky--Saleur and Mikhaylov. In particular, we match Verlinde formulae and
mapping class group actions on state spaces of non-generic tori with results in
the physics literature. We also obtain explicit descriptions of state spaces of
generic surfaces, including their graded dimensions, which go beyond results in
the physics literature.
