We describe how to form a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which performs a crepant resolution. In addition we describe a possible mirror construction, which allows us to describe the Picard-Fuchs operators associated to these examples. We focus particularly on a class of examples related to joins of elliptic curves, including examples recently considered by Inoue and Knapp-Sharpe.