Right now, you may be locked into mowing as the heads emerge. 

However, in the spring when the downy brome heads first emerge, you have time from the time the heads emerge, until the seeds viable.   In the spring, I expect that you don't seed heads emerge all at the same time.  That is where I would seed heads emerge, and then wait a week to allow most- if not all - heads emerge.  

What I suspect is happening is that you start to mow when you start to seed heads emerge.  Once the seed heads emerge, those particular stems are done; however, there are also other stems - tillers - that are still in "boot" stage - meaning the heads are still down inside the stem.  These are the stems that try to regrow, and are causing you problems.  

From the time the seeds heads emerge, I think it will take 2-3 weeks for the seeds to become manure. You want to fit your mowing within this window.  I thinking waiting a week after seed head emergence means the seeds are not yet mature and should give you results you need.

In the fall, learn how to identify cheat grass/ downy brome when it is first germinating. I would start looking in mid-October, and periodically until you see it is germinating and emerging.  The leaves are hairy, and I think it has a different "look" compared to other grasses.  If you can identify it in the fall, then you know when you use an herbicide.