It appears that what you're looking for is a retail (full) digital license for Windows 11 Pro, because that's the only type Microsoft allows you to transfer to new hardware. OEM/System Builder keys (the cheap ones you often see online) are permanently tied to the first motherboard and won't solve your problem.
You can buy a digital Windows 11 Pro license directly through Windows itself. Open Settings, go to System, then Activation, and choose “Upgrade your edition of Windows” or “Open Store.” That takes you to the Microsoft Store page where you can purchase Windows 11 Pro digitally. After purchase, the license is tied to your Microsoft account and activates automatically. This is a retail license and is transferable.
If that path doesn't show the purchase option, you can also buy it from Microsoft's website. Go to microsoft.com, search for “Windows 11 Pro,” and choose the digital download version (not USB or physical). As long as it's sold directly by Microsoft and not labeled OEM, it's a retail license and transferable.
Given your situation, you may not actually need to buy a new license yet. A retail Windows 10 Pro key should still activate Windows 11 Pro, even after a motherboard change, but it often requires manual reactivation. The key step is using the activation troubleshooter while signed into the Microsoft account that originally held the license. Go to Settings > System > Activation and run the troubleshooter, then select “I changed hardware on this device recently” and reassign the license. If your original key was truly retail, this usually works even after major upgrades.
If that fails, you can also try activating directly with your old key using:
slmgr /ipk YOUR-WINDOWS-10-PRO-KEY
slmgr /ato
If those methods don't work, it's possible your original license wasn’t actually retail (many “retail” keys sold online are OEM or volume licenses), which would explain why it won't transfer.
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hth
Marcin