That's one possible approach, and one possible outcome, but not the only one of either.

For instance, rather than "proving" a pattern of pay discrimination, she might establish other facts that demonstrate that her lower pay was based on her sex. If a supervisor makes a remark about "having to hire a woman" or giving her "the girl jobs" to do, that could be evidence of bias.

The EEOC is unlikely to harass a company over a baseless claim. If she does handle it badly and ruin her reputation, changing fields would not be likely to help - unless her field has a bias against women, thinking that they are a lawsuit risk.

The most likely outcome, if she approaches the company correctly, is that they will provide some valid explanation for the difference, such as their respective grades in their classwork. It would be interesting to hear the actual results, though.