  Given a degenerate Calabi-Yau variety $X$ equipped with local deformation
data, we construct an almost differential graded Batalin-Vilkovisky (dgBV)
algebra $PV^{*,*}(X)$, producing a singular version of the extended
Kodaira-Spencer differential graded Lie algebra (dgLa) in the Calabi-Yau
setting. Assuming Hodge-to-de Rham degeneracy and a local condition that
guarantees freeness of the Hodge bundle, we prove a
Bogomolov-Tian-Todorov--type unobstructedness theorem for smoothing of singular
Calabi-Yau varieties. In particular, this provides a unified proof for the
existence of smoothing of both $d$-semistable log smooth Calabi-Yau varieties
(as studied by Friedman and Kawamata-Namikawa and maximally degenerate
Calabi-Yau varieties (as studied by Kontsevich-Soibelman and Gross-Siebert). We
also demonstrate how our construction yields a logarithmic Frobenius manifold
structure on a formal neighborhood of $X$ in the extended moduli space by
applying the technique of Barannikov-Kontsevich.
