  Let $\mathcal{L}$ be a pencil of plane curves defined over $\mathbb{F}_q$
with no $\mathbb{F}_q$-points in its base locus. We investigate the number of
curves in $\mathcal{L}$ whose $\mathbb{F}_q$-points form a blocking set. When
the degree of the pencil is allowed to grow with respect to $q$, we show that
the geometric problem can be translated into a purely combinatorial problem
about disjoint blocking sets. We also study the same problem when the degree of
the pencil is fixed.
