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People who have direct experience of such things say that recourse to cats is made to disrupt people who are getting too involved in debate. In the interest of reminding you that nothing is new, I quote Thomas Cleary's translation of case 9 in ‘The Book of Serenity’, which goes slightly further than the judgement of Solomon:

One day at Nanquan's the eastern and western halls were arguing over a cat. When Nanquan saw this, he took and held it up and said, “If you can speak I won't cut it.”

The group had no reply; Nanquan then cut the cat in two.

Nanquan also brought up the foregoing incident to Zhaozhou and asked him: Zhaozhou immediately took off his sandals, put them on his head, and left. Nanquan said, “If you had been there you could have saved the cat.”

Wansong, the original author, explains in his commentary of this case.

Although this little bit of activity is difficult to understand, yet it is easy to see—just see through it in lifting the spoon and picking up chopsticks, and you will see that the cutting of the cat and the wearing of the sandals on the head are not any different.

He doesn't mention whether Nanquan goes on to ask, “I can has enlightenment?”