Neither of
- "After being broken up with my boyfriend"
- "After broken up with my boyfriend"
is correct English. "To break up with someone" is to end a romantic and/or sexual relationship. (More rarely it can mean to end a non-romantic friendship.) One does not say "being broken up with" but rather "having broken up with" or "I broke up with" to indicate that this happened in the past. "break up with" is a verb phrase in this usage. It would not be used with "being" in any tense. For a construction with "being" one would use an -ed form, such as "being separated from"
Without context I cannot be sure, but I suspect the original was simply an informal, incorrect usage. It is not a common enough usage to become correct by virtue of usage.