What is Draw Me A Kitten?
1.
"Draw me a kitten" is reference to any menial task that a supervisor or higher-up assigned for you to perform. Basically something to keep you busy until some "real" work comes along.
"Uggh, Bob's got me drawing kittens all day for him."
OR
(imitating your boss) "Blah, blah, blah, I need you to draw me a kitten, Mike.....and I want it cute! And name it 'Mittens!' And draw a scratching post!"
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1. 
A fucked up, but very funny song by Adam Sandler from his 1993 album They're All Gonna Laugh at You!.
Lyrics:
Put your arms aroun..