FireAndForgetTest.cs

/*
  SharpLab tools in Run mode:
    • value.Inspect()
    • Inspect.Heap(object)
    • Inspect.Stack(value)
    • Inspect.MemoryGraph(value1, value2, …)
*/
using System;
using System.Threading.Tasks;

Console.WriteLine("🌄");


//DoSomething_Awating().FireAndForget(e => Console.WriteLine(e.Message));
DoSomething_ReturningTask().FireAndForget(e => Console.WriteLine(e.Message));

await Task.Delay(3000);


static Task DoSomething_ReturningTask()
{
    throw new Exception("My one");
    return Task.Delay(2000);
}


static async Task DoSomething_Awating()
{
    throw new Exception("My one");
    await Task.Delay(500);
}

static class TaskEx
{
    public static void FireAndForget(this Task task, Action<Exception>? errorHandler = null)
    {
        task.ContinueWith(t =>
                          {
                              Console.WriteLine(t.Exception.GetType());
                              if (t.IsFaulted && errorHandler != null)
                                  errorHandler(t.Exception.InnerException);
                              
                          }, TaskContinuationOptions.OnlyOnFaulted);
    }
}

you can run that code from here