NamedCredentailsExample is in Source org with named credentials to hit target salesforce org. NamedCredentailsLogCapture is in target org to handle incoming requests

NamedCredentailsExample.cls

public class NamedCredentailsExample {

    public static String oAuth_flow(){
        return makeCallout('OAuth_flow_Demo/services/apexrest/namedcredentials/demo','GET', null,null);
    }
    
     public static String oAuth_flow_Post(String body){
        return makeCallout('OAuth_flow_Demo/services/apexrest/namedcredentials/demo','POST', null,body);
    }
    
    public static String salesforce_password_flow(){
        Map<String, String> headers = new Map<String, String>{'X-Username' =>'targetusername', 'X-Password' => 'targetpassword'};
        return makeCallout('Salesforce_Password_Flow_Demo/services/apexrest/namedcredentails/demo', 'GET',headers,null);
    }
    
    public static String password_flow(String body){
        return makeCallout('Password_Flow_Demo/namedcredentails/demo', 'POST', null, body);
    }


    private static String makeCallout(String namedcredentails, String method, Map<String, String>headers, String body){
        HttpRequest req = new HttpRequest();
        req.setEndpoint('callout:'+namedcredentails);
        
        if(headers != null){
            for(String key : headers.keySet()){
                req.setHeader(key, headers.get(key));
            }
        }
        
        // req.setHeader('Authorization', 'OAuth {!$Credential.OAuthToken}');
        req.setHeader('Content-Type', 'application/json');
        req.setMethod(method);
        
        req.setBody(body);
        Http http = new Http();
        HTTPResponse res = http.send(req);
        return res.getBody();
    }
}

NamedCredentailsLogCapture.cls

@RestResource(urlMapping='/namedcredentials/demo')
global class NamedCredentailsLogCapture {

  @HttpGet  
  global static String getInformation() {
    return UserInfo.getUserName();
  }

  @HttpPost  
  global static String captureLogs() {
    RestRequest req = RestContext.request;
    try{
      NamedCredentailsLogCapture.addLog(String.valueOf(req.headers), req.requestBody.toString());
    }catch(Exception e){

    }
    return null;
  }


  public static void addLog(String headers, String body){
    Named_Credentails_Log__c log = new Named_Credentails_Log__c();
    log.Body__c = body;
    log.Headers__c = headers;
    insert log;
  }
}

Named credentials sample code