var timer = new System.Timers.Timer(TimeSpan.FromSeconds(1).TotalMilliseconds)
{
  AutoReset = true,
  Enabled = false
};
timer.Elapsed += TimerElapsed;
timer.Start();
...
static async void TimerElapsed(object sender, System.Timers.ElapsedEventArgs e)
{
  await Task.Delay(TimeSpan.FromSeconds(1));
  Console.WriteLine("Tick: {0}", DateTime.Now);
}

static async Task Main()
{
  AppDomain.CurrentDomain.UnhandledException += (s, e)
    => Console.WriteLine("UnhandledException: {0}", e.ExceptionObject);
  TaskScheduler.UnobservedTaskException += (s, e) =>
    Console.WriteLine("UnobservedTaskException: {0}", e);
  Console.WriteLine("Running TestTask");
  try
  {
    await TestTask();
  }
  catch (Exception e)
  {
    Console.WriteLine("TestTask error: {0}", e);
  }
  Console.WriteLine("Running TestVoid");
  try
  {
    TestVoid(); // здесь await использовать нельзя
  }
  catch (Exception e)
  {
    Console.WriteLine("TestVoid error: {0}", e);
  }
  Console.WriteLine("done");
}

static async void TestVoid()
{
  await Task.Yield();
  throw new ApplicationException("TestVoid");
}

static async Task TestTask()
{
  await Task.Yield();
  throw new ApplicationException("TestVoid");
}

Running TestTask
TestTask error: System.ApplicationException: TestVoid
   at test2.Program.<TestTask>d__2.MoveNext() in D:\Projects\.Net\_tests\test2\test2\Program.cs:line 54
--- End of stack trace from previous location where exception was thrown ---
   at System.Runtime.ExceptionServices.ExceptionDispatchInfo.Throw()
   at System.Runtime.CompilerServices.TaskAwaiter.HandleNonSuccessAndDebuggerNotification(Task task)
   at System.Runtime.CompilerServices.TaskAwaiter.GetResult()
   at test2.Program.<Main>d__0.MoveNext() in D:\Projects\.Net\_tests\test2\test2\Program.cs:line 26
Running TestVoid
done
UnhandledException: System.ApplicationException: TestVoid
   at test2.Program.<TestVoid>d__1.MoveNext() in D:\Projects\.Net\_tests\test2\test2\Program.cs:line 48
--- End of stack trace from previous location where exception was thrown ---
   at System.Runtime.ExceptionServices.ExceptionDispatchInfo.Throw()
   at System.Threading.ExecutionContext.RunInternal(ExecutionContext executionContext, ContextCallback callback,
        Object state, Boolean preserveSyncCtx
)
   at System.Threading.ExecutionContext.Run(ExecutionContext executionContext, ContextCallback callback,
       Object state, Boolean preserveSyncCtx)
   at System.Threading.QueueUserWorkItemCallback.System.Threading.IThreadPoolWorkItem.ExecuteWorkItem()
   at System.Threading.ThreadPoolWorkQueue.Dispatch()

Unhandled Exception: System.ApplicationException: TestVoid
   at test2.Program.<TestVoid>d__1.MoveNext() in D:\Projects\.Net\_tests\test2\test2\Program.cs:line 48
--- End of stack trace from previous location where exception was thrown ---
   at System.Runtime.ExceptionServices.ExceptionDispatchInfo.Throw()
   at System.Threading.ExecutionContext.RunInternal(ExecutionContext executionContext, ContextCallback callback,
       Object state, Boolean preserveSyncCtx
)
   at System.Threading.ExecutionContext.Run(ExecutionContext executionContext, ContextCallback callback,
      Object state, Boolean preserveSyncCtx)
   at System.Threading.QueueUserWorkItemCallback.System.Threading.IThreadPoolWorkItem.ExecuteWorkItem()
   at System.Threading.ThreadPoolWorkQueue.Dispatch()