I don’t think it works that way, and if it does, why hasn’t he?
I admit it has never been done. But I just re-read the appropriate sections of the Senate Rules, and there are no restrictions on the VP doing this.
In fact, there is one section that says the Presiding Officer can Adjourn or Recess. This was done in 1994 in the middle of a Quorum Call by the Presiding Officer. Remember, the VP is the Presiding Officer of the U. S. Senate.