We will discuss how the Fourier-Mukai transform acts on sheaves supported on the closed stratum of the singular locus of the compactified Jacobian of an irreducible nodal curve, by relating it to the classical transform on the Jacobian of its normalisation. We will then explain how this can be applied to the study of mirror symmetry for certain Lagrangian subvarieties - more precisely, BAA-branes - in moduli spaces of Higgs bundles lying over the singular locus of the Hitchin fibration h. We will see that mirror symmetry for these branes exhibits features that differ from the better-understood case of branes lying over the regular locus of h. This is joint work in progress with E. Franco, R. Hanson, and J. Horn.