MyAsyncWebServer prototype. Some code errors inside.

CallbackTest.ino

#include <ESPAsyncTCP.h>
#include <Ticker.h>
#include <ESPAsyncWebServer.h>
#include <ESP8266WiFi.h>
#include <functional>
#include <Hash.h>

using namespace std;
using namespace placeholders;

class MyServer : public AsyncWebServer {
public:

  MyServer(int port) : AsyncWebServer(port) {
  } 

  void begin() {
    _periodicTrigger.attach(10.0f, &MyServer::s_periodicTask, static_cast<void*>(this));

    on("/index.html", std::bind(&MyServer::indexHandler, this, _1));
	
	on("/", [](AsyncWebServerRequest *request) {
		AsyncWebServerResponse *response = request->beginResponse(302);
		response->addHeader("Location", "/index.html");
		response->addHeader("Cache-Control","no-cache");
		request->send(response);
	});

	onNotFound([](AsyncWebServerRequest *request) {
		request->send(404,"text/plain","Not Found");
	});

    AsyncWebServer::begin();
  }
protected:
  Ticker _periodicTrigger;
  int value = 0;


  void indexHandler(AsyncWebServerRequest* request) {
	  char* text = (char*)malloc(20 * sizeof(char));
	  sprintf(text, "Value is %d", value);
	  Serial.printf("%s\r\n", text);
	  request->send(200,"text/plain",text);
	  delete(text);
  }

  void periodicTask() {
    Serial.printf("Periodic task. Result = %d\n", value);
    value++;
  }

  static void s_periodicTask(void* arg) {
    MyServer* self = reinterpret_cast<MyServer*>(arg);
    self->periodicTask();
  }

};

MyServer server(80);

void additionalHandler(AsyncWebServerRequest* request) {
  request->send(200, "text/plain", "normal handler");
}

void setup()
{
  Serial.begin(115200);
  WiFi.begin("rohde", "RS0MADRID28160869");
  server.on("/other.html", additionalHandler);
  server.begin();
  Serial.println(ESP.getFreeSketchSpace());
}

void loop()
{
}

periodicTask takes one argument, so you need one placeholder: std::bind(&MyServer::indexHandler, this, std::placeholders::_1). For two arguments you add a 4th parameter std::placeholders::_2 and so on.

And make the base class public: class MyServer : public AsyncWebServer