  Constraining the vertical and radial structure of debris discs is crucial to
understanding their formation, evolution and dynamics. To measure both the
radial and vertical structure, a disc must be sufficiently inclined. However,
if a disc is too close to edge-on, deprojecting its emission becomes
non-trivial. In this paper we show how Frankenstein, a non-parametric tool to
extract the radial brightness profile of circumstellar discs, can be used to
deproject their emission at any inclination as long as they are optically thin
and axisymmetric. Furthermore, we extend Frankenstein to account for the
vertical thickness of an optically thin disc ($H(r)$) and show how it can be
constrained by sampling its posterior probability distribution and assuming a
functional form (e.g. constant $h=H/r$), while fitting the radial profile
non-parametrically. We use this new method to determine the radial and vertical
structure of 16 highly inclined debris discs observed by ALMA. We find a wide
range of vertical aspect ratios, $h$, ranging from $0.020\pm0.002$ (AU Mic) to
$0.20\pm0.03$ (HD 110058), which are consistent with parametric models. We find
a tentative correlation between $h$ and the disc fractional width, as expected
if wide discs were more stirred. Assuming discs are self-stirred, the thinnest
discs would require the presence of at least 500 km-sized planetesimals. The
thickest discs would likely require the presence of planets. We also recover
previously inferred and new radial structures, including a potential gap in the
radial distribution of HD 61005. Finally, our new extension of Frankenstein
also allows constraining how $h$ varies as a function of radius, which we test
on 49 Ceti, finding that $h$ is consistent with being constant.
