So to make sure I'm tracking:

This is the outlet I need to install:
https://www.homedepot.com/p/Leviton-30-Amp-2-Pole-Flush-Mount-Shallow-Single-Outlet-Black-R10-05372-S00/300324202

This is the breaker I need to put in the panel:
https://www.homedepot.com/p/Eaton-CH-30-Amp-2-Pole-Circuit-Breaker-with-Trip-Flag-CHF230/204223356

For the 10/2 - both white and black go to the double pole breaker, brown to ground. https://www.amazon.com/gp/product/B000BQS5TG/?th=1

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This is the compressor I own that I'm trying to hook up - https://www.amazon.com/gp/product/B00TDNKBKE/

This is the welder/cutter I have - https://yeswelder.com/products/firstess-mp200

Thanks!