For which $f \in \mathbb{C}[x,y]$, $fy-\lambda$ is irreducible in $\mathbb{C}[x,y]$ for infinitely many $\mathbb{C} \ni \lambda$'s

Question

Let $f=f(x,y) \in \mathbb{C}[x,y]$, and for every $\lambda \in \mathbb{C}$, denote $F_{\lambda}:=fy-\lambda$.

Is it possible to characterize all $f \in \mathbb{C}[x,y]$ such that $F_{\lambda}$ is irreducible in $\mathbb{C}[x,y]$ for infinitely many $\mathbb{C} \ni \lambda$'s?

Example: If I am not wrong, $f=x$ is such that $F_{\lambda}=xy-\lambda$ is irreducible for every $\lambda \in \mathbb{C}-\{0\}$.

See also this question concerning irreducibility of a polynomial in two variables.

Any hints and comments are welcome!