SSL client verification required

I'm confused about how to implement SSL/TLS for connecting 2 servers so here is my scenario.

 

Server A Obligations:

 

the server listens for tcp(non http) traffic on port 443.

 

it will create a self sign private + public.

 

it will send manually the public for Server B.

 

Client verification is required.

 

Server B obligations:

 

should establish the connection and keep it open and in case o disconnection should establish the connection again

 

should create self sign private and public

 

should send manually the public for Server A.

 

Client verification is required I should mention that F5 LTM will be between the servers and should offload the burden of Encryption and Decryption for server B