I have no idea whether this type of “joke” has a technical term for it, but I do remember that I used to sit for hours trying to make them up, both as a child and as a student.
For some reason I dreamed about this form last night and, in the dream, people referred to them as quizmaster jokes, which seems as good a name as any.
I also thought of one in my sleep and, of course, I then sat and thought up two more when I woke.
The one I dreamed
What did the quizmaster say when asked whether he would prefer to spend the afternoon bandaging his lower leg or hunting for his long-lost shoe?
He said “Frankly, I would rather bind my foot than find my boot”.
And then we have
What did the quizmaster say when asked whether he would prefer to communicate with his nemesis in writing or repair his yacht with a needle and some yarn?
He said, “The former. I would much rather send a mail than mend a sail”.
Finally, for the moment at least
What did the quizmaster say when asked whether he would prefer to conduct a church service for a bunch of recently deceased idiots or pay far too much attention to a group of nearby donkeys?
He said, “I would rather fawn over mules than mourn over fools”.
The last one only works if you say it out loud. It won’t work on the page, as the others do.
On the other hand all of them demonstrate the perils of staying indoors for too long.