There have been a few results of this type published, although I am struggling to remember where to find them right now.

There is a paper of Glibichuk and Rudnev (here is an arxiv version, and here is the published version) which shows that, for any $A,B \subset \mathbb F_q$
$$
|A||B| \geq q \Rightarrow 10A \cdot B = \mathbb F_q.
$$
Here $10A \cdot B=A\cdot B+A\cdot B+\dots+A\cdot B$, as in your comment above. I am pretty sure there is a paper of Iosevich and Rudnev (and possibly other authors) which proves that
$$
|A| \gg q^{3/4} \Rightarrow AA+AA=\mathbb F_q.
$$
Or something close to this statement. Perhaps someone else will be able to dig out the reference.

In Theorem 30 of this long paper, it was proven that, for $A \subset \mathbb F_p$ with $p$ prime,
$$
|A| \gg p^{3/5} \Rightarrow (A-A)(A-A)(A-A)(A-A)= \mathbb F_p.
$$

I think there are many other variants of these kinds of results out there. A related and somewhat easier question is to consider the threshold for which these expanders are guaranteed to generate a positive proportion of the elements of the finite field. But this answer is already too long.