encryption – How can I encrypt something with a PGP private key?

I understand the normal operation is to encrypt with someone’s public key and decrypt with the private. And I’m also aware of how signatures usually work. However I’d like to ensure that in order to access the data, the recipient has actually gotten my public key and decrypted the data with it. My understanding this is possible with RSA. Is that correct that this is possible?

If so, how do I do this? And how do I then decrypt the resulting file? Ideally this is something that GPG would be able to do (tho it doesn’t look like Kleopatra on windows is able to do that that I can see).