A silly comment - since $X$ is a symplectic specialization of $Z$, $\check{X}$ should be complex specialization of $\check{Z}$. So you will need to understand how to extract a semi-orthogonal decomposition of the Fukaya-Seidel category from a specialization family. For example, the mirror of a del Pezzo $dP_{n}$ is a rational elliptic surface with an $I_{9-n}$ fiber at infinity. The blow-up of $dP_{n}$ at a point corresponds to morsifying one node of the fiber at infinity, i.e. to a splitting it as $I_{1}+I_{9-(n+1)}$ and morsification gives you a semi-orthogonal decomposition of FS.