Let me guess, you ordered the computer from an OEM and they shipped you Win 11 Home. OEM versions of Windows are hard coded to the motherboard these days. That means that whenever you restore it'll go back to the Win 11 Home license that came with the system.
If you already have an existing OS license for Windows 10 Pro (and you have a free upgrade to Win 11) then you'll need to install your copy of Win 10 Pro on the new machine (don't use the OEM restore). Then you can upgrade to Win 11 Pro given the free MS upgrade offer. You won't be using the Win 11 Home OEM license at all.
Note also that MS switched to tying your licenses to your MS account so assuming that you already tied your Win 10 Pro license to your MS account then when you install Win 10 Pro and then link your MS account it should automatically find the license and you shouldn't need a product key anymore. Therefore don't enter a product key when you first install Win 10.