Suppose $X \subset Y$ is a pair of varieties, and $s \in H^0(N_{X/Y})$ is a section. This corresponds to a first-order deformation $X' \subset Y \times \text{Spec}(\mathbb{C}[\epsilon]/\epsilon^2)$ of $X \subset Y$.

If $\mathcal{E} \to Y$ is a vector bundle, is there a nice way to compute the cohomology of the restriction of $\mathcal{E}$ to the generic fiber of $X'$ (in terms of $X \subset Y$, $\mathcal{E}$, and $s$)?

EDIT: Oops, I meant to write "the restriction of $\mathcal{E}$ to $X'$" (not to its generic fiber, which doesn't make sense as pointed out below).