  We develop a Floer theoretical gluing technique and apply it to deal with the
most generic singular fiber in the SYZ program, namely the product of a torus
with the immersed two-sphere with a single nodal self-intersection. As an
application, we construct immersed Lagrangians in $\mathrm{Gr}(2,\mathbb{C}^n)$
and $\mathrm{OG}(1,\mathbb{C}^5)$ and derive their SYZ mirrors. It recovers the
Lie theoretical mirrors constructed by Rietsch. It also gives an effective way
to compute stable disks (with non-trivial obstructions) bounded by immersed
Lagrangians.
