encryption – Check signature of an HTTPS response

So I want to do the following

1. Check the RSA public key of a https server
2. Make a GET request to that server
3. Get a response+signature from the server
4. I should be able to check if ANY reponse+signature pair matches the publickey.

I was able to get the public key based on another qeustion https://security.stackexchange.com/a/16091

openssl s_client -connect the.host.name:443 | openssl x509 -pubkey -noout

But I don’t know how to get a signature on the response message.
Is it possible? Do I need to specify a request header with curl? Am I missing something?

The reason I need this is because I’m trying to setup a smart contract (blockchain), with the public key stored. People will send messages and I want to be able to check the messages come from server.