  We explain how to form a novel dataset of simply connected Calabi-Yau
threefolds via the Gross-Siebert algorithm. We expect these to degenerate to
Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily
isolated) singularities. In particular, we explain how to `smooth the boundary'
of a class of $4$-dimensional reflexive polytopes to obtain a polarised
tropical manifolds. We compute topological invariants of a compactified torus
fibration over each such tropical manifold, expected to be homotopy equivalent
to the general fibre of the Gross-Siebert smoothing. We consider a family of
examples related to the joins of elliptic curves. Among these we find $14$
topological types with $b_2=1$ which do not appear in existing lists of known
rank one Calabi-Yau threefolds.
