You can say "she broke in" not with a me, as in While I was speaking she broke in and disputed my findings. It's an abreviation of broke in on/to the conversation but interrupted is more commonly used

Nice guess, given the literal translation of “interrupt” from the Latin. You could say “she broke in on me” or “she broke in on my monolog.” Be careful, though, as as the version without “on” has no meaning. As a separable phrasal verb, to “break in” exists, but “she broke me in” has an entirely different meaning. It would be best just to say “she interrupted me”.

No, you cannot. Because "broke in" means "ворваться (куда-нибудь)". "interrupt" means "прервать (кого-то)/перебить".