Can a sprocket be re-cut?

It would be a fairly simple thing to make, it would just take time,,, but any machine shop is going to have to charge you way more than you'd spend buying a off-the-shelf sprocket.
Around where I live, manual shops charge around $100/hr.
CNC shops charge at least 3X or 4X that much.
Yes, a CNC shop could rip it out in an hour probably, but still.

What's a lot more cost-effective would be to find a "solid/disk-style" existing BMX sprocket already made that will work, and have a manual machine shop drill new mounting holes in it for you. They might offer a lower price if you are willing to leave it there for a week or two so they can do it in their spare time, rather than interrupt their "bread and butter" work.
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Looks to me like Bandito and Bairdo are the only two with common sense. For 10-20 bucks you can get a 36t HAPPY TIME sprocket from guys like BGF. I picked up a few last year from him for $12 ea. with mounting hardware. Silly to consider machining a BMX or 44t original. I did this once with indexer and drill press. NOT worth the effort.

Actually it's a sprocket off a Sparta moped so it was made by the Dutch.