Depends upon if you still want to overseed with the braccs.
If so, mow the plots now, wait two weeks and then spray the new growth. Give these a couple of weeks to die, then overseed, that should put you into early August.
The deer will still be getting some use from the plots after the weeds are cut and start to regrow the two weeks after mowing, and the two weeks it's going to take for the herbicide to really kill things.
“There are two spiritual dangers in not owning a farm. One is the danger of supposing that breakfast comes from the grocery, and the other that heat comes from the furnace.”