If you don't want the task started, just use new Task as you were using, but then you need to call Start() method on that task before awaiting it!
[Reference]
My recommendation
Just make a method that returns the anonymous function, like this:
private static Func
Then get it and run it in a new Task whenever you need it (Also notice that I removed the async/await from the lambda expression, since you are putting it into a Task already):
Task.Factory.StartNew(GetFunction());
One advantage to this is that you can also call it without putting it into a Task: