C++ overload template with specific signature

I have the following:

struct Args;

template <typename T>
void DoStuff(T&& callback) {
    // ... do stuff
    MyArgs args = ...
    callback(args);
}

Which is great, I can do the following:

void callback(const Args &);
DoStuff(callback);

And:

DoStuff([](const Args &) { ... });

And:

class CallbackClass {
    operator()(const Args &);
};
CallbackClass myCallbackClass;
DoStuff(myCallbackClass);

All is nice and good, now I want to do two improvements here:

1. Enforce that the signature of T is always void(const Args &) because in the current state of things I can do the following:

   void badCallback(Args);
   DoStuff(badCallback);
   

2. Allow (in addition to the previous options) to pass an object with an specific named member function that would also be allowed to be a callback, example:

   class NamedCallback {
       void PerformCallback(const Args &);
   };
   NamedCallback myNamedCallback;
   DoStuff(myNamedCallback);
   

Is it possible?