Enforce an async method to be called once

I'd go with AsyncLazy<T> (slightly modified version):

public class AsyncLazy<T> : Lazy<Task<T>> 
{ 
    public AsyncLazy(Func<T> valueFactory) : 
        base(() => Task.Run(valueFactory)) { }

    public AsyncLazy(Func<Task<T>> taskFactory) : 
        base(() => Task.Run(() => taskFactory())) { } 

    public TaskAwaiter<T> GetAwaiter() { return Value.GetAwaiter(); } 
}

And consume it like this:

private AsyncLazy<bool> asyncLazy = new AsyncLazy<bool>(async () =>
                                    { 
                                        await DoStuffOnlyOnceAsync()
                                        return true;
                                    });

Note i'm using bool simply because you have no return type from DoStuffOnlyOnceAsync.

Edit:

Stephan Cleary (of course) also has an implementation of this here.

Yes. Use Stephen Cleary's AsyncLazy (available on the AsyncEx nuget):

private static readonly AsyncLazy<MyResource> myResource = new AsyncLazy<MyResource>(
    async () => 
    { 
        var ret = new MyResource(); 
        await ret.InitAsync(); 
        return ret; 
    }
);

public async Task UseResource()
{
    MyResource resource = await myResource;
    // ...
}

Or the visual studio SDK's AsyncLazy if you prefer a Microsoft implementation.

I have a blog post that covers a few different options for doing "asynchronous constructors".

Normally, I prefer asynchronous factory methods, because I think they're simpler and a bit safer:

public class MyService
{
  private MyService() { }

  public static async Task<MyService> CreateAsync()
  {
    var result = new MyService();
    result.Value = await ...;
    return result;
  }
}

AsyncLazy<T> is a perfectly good way of defining a shared asynchronous resource (and may be a better conceptual match for a "service", depending on how it is used). The one advantage of the async factory method approach is that it's not possible to create an uninitialized version of MyService.