*   How would we be able to achieve this? I have read somewhere on a 2016 
mail in the archives that it is an "encrypt-only" algorithm  and that is not 
possible.

X25519 is a key-exchange-only algorithm.  Ed25519 is a signing algorithm.  
Unlike classic RSA, the signing and the key exchange are two different 
operations (well, technically RSA doesn’t have key exchange).  Both are defined 
by IETF RFC’s.  OpenSSL doesn’t fully support Ed25519.



  *   But I have found many sites on let's encrypt already using this.

Are you sure?  Please post a key.  Ed25519 is quite different from EdDSA or 
ECDSA or DSA, which typically use a P-256 curve.

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