Yes, the noun phrase leave of absence is countable, with the plural being leaves of absence. This is confirmed by the following dictionary example, despite the entry marking the head noun leave as a mass noun.
leave noun2 (mass noun)
1 Time when one has permission to be absent from work or from duty in the armed forces.
‘Full-time officers of research may take leaves of absence according to the policies described below.’
Suppose one goes on leave for medical reasons, and upon the period's expiry takes personal leave, they are on leave (non-count) for the whole duration. They aren't on (*) two leaves. Likewise, if another person takes some time off as well, we still say that the two people are on leave (still non-count). It's non-count because it refers to a state of their employment. There is no change to the state even if multiple episodes are ganged together.
Since on leave refers to state, it doesn't become plural even when talking about separate, non-consecutive episodes. One says that someone has been on leave twice, not that they have been on (*) two leaves.
However, a leave of absence is understood as a defined episode:
A leave of absence (LOA) is a period of time that one must be away from one's primary job, while maintaining the status of employee. This contrasts with normal periods away from the workplace, such as vacations, holidays, hiatuses, sabbaticals, and "working from home" programs, in that they are considered exceptional circumstances, rather than benefits. Generally such an arrangement has a predefined termination at a particular date or after a certain event has occurred.
Because these episodes can be counted, the term leave of absence is treated as countable.
(*) The asterisk is a convention that ELU uses to flag instances of non-grammatical usage.