Joshua Finch wrote: They will resprout, but cut them again at the same growth density. Do this all summer and gain valuable mulch material.
Then you best be ready to seed with what you have planned, or else other more undesirable species are going to fill those gaps. That is for certain.
Still, I would imagine that if you are working for an herb company that it wouldn't be so difficult to continue chop and dropping where they are near plants (if you don't sheet mulch somehow). I wouldn't say this is a problem.