I don't have a good answer for you, because I use a Silpat mat from Williams-Sonoma, and it washes easily. Eventually, it gets a little grimy, as do the cookie sheets, and then I throw them all away and go buy new ones. Sorry I'm not more help!!!
I 2nd (or is it 3rd) the parchment paper. You'll never toss another cookies sheet again. Even better, they won't require cleaning each and every time and therefore won't develope rust spots so easily or quickly.
I use parchment paper on my cookie sheets, it is wonderful to work with. Also, if you use a baking stone instead of cookie sheets, your cookies turn out great and the clean up is so so easy!
I find that residue worse when I spray pans with cooking spray. So I save two non-stick pans that I won't spray. I should use parchment paper more often.
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