I imagine him writing a letter with something like the following text.
Gentlemen,
I am required by act of Parliament to ask you for an extension of the October 31 deadline for the U.K. to leave the EU. Consider this letter to constitute that request.
Please understand that I sincerely hope you will refuse this request and allow the U.K. to leave on 31 October. I have little faith that any agreement you might accede to could muster a majority in Parliament, however long the leaving is delayed.
It is my hope you find the limbo-like current situation as distasteful as do I. It is time the waiting ends and we get on with discovering together the nature of our future relations. These I hope and believe can be amicable and mutually profitable, if configured somewhat differently than at present.
Sincerely,
The Right Honorable Boris Johnson,
Prime Minister of Her Majesty’s Government.