Let $X$ be a smooth, projective variety over $\mathbb{C}$ for which $\chi(X) = 0$. Here by $\chi$, I mean the topological Euler characteristic of $X(\mathbb{C})$; this number can also be computed as the degree of the top Chern class of the tangent bundle of $X$, and there are various other equivalent definitions.

I want to find/construct smooth hyperplane sections $Y$ of $X$ for which $\chi(Y) \neq 0$; I do not care too much about the projective embedding. Under which additional conditions (if any) is doing such a thing possible or reasonable? What are potential obstructions, et cetera?

Motivation: I can prove an interesting statement for varieties with $\chi \neq 0$. I believe it to be true for varieties with $\chi = 0$ as well, but alas my arguments do no longer apply. I want to try to reduce it to the case $\chi \neq 0$ using suitably chosen hyperplane sections.