Non-blocking dispatcher thread with Control and System.Threading.Thread

Question

The problem is that dispatcher blocks the MainWindow thread until it is executed all of the queued AddOutputLine()s. Can these calls be non-blocking for a wpf control?

    //TaskPage.xaml.cs
    [MTAThread]
    public void AddOutputLine(string line = "")
    {
        try {
            Session.TaskPage.Dispatcher.BeginInvoke(new Action(delegate
            {
                txtOutput.Text += "[" + DateTime.Now + "] " + line + Environment.NewLine;
                txtOutput.ScrollToEnd();
            }));
        } catch (Exception ex) {
            Program.SetCurrentStatus("Error occured : "+ex.Message);
        }
    }

In this class that invokes the Action parameter:

//BackgroundUploader.cs
private Action<string> _log;
public BackgroundUploader(Client client, Action<string> log = null)
{
    _log = log;
    _client = client;
}

while doing this:

    //BackgroundUploader.cs
    public void RunThreadProc()
    {
        try
        {
            _log?.Invoke("Timeout interval set for BackGround Task.");
            while (!Stop)
            {
                //the problematic blocking of WPF Control starts here
                var inOutFiles = CreateXML(_sqlStrings, _outputNames, _outputDirectory);
                Send(inOutFiles, _outputDirectory);
                LastRunTime = DateTime.Now;
                _log?.Invoke($"Waiting for {_waitMinutes} minutes...");
                //the Control returns to processing messages
                System.Threading.Thread.Sleep((int)TimeSpan.FromMinutes(_waitMinutes).TotalMilliseconds);
            }
            _log?.Invoke("BackGroundTask canceled.");
        }
        catch (Exception ex)
        {
            _log?.Invoke($"Error while executing background task: {ex.Message}");
        }
    }

The RunThreadProc() is called from Control like this:

//TaskPage.xaml.cs
_backgroundThread = new System.Threading.Thread(bU.RunThreadProc);
_backGroundThread.Start();

I also cannot use Bcl.Async because it's failing on Task result with method that uses async modifier on Windows Server 2003 target OS.

edit: I encountered the problem inside of the background class, its methods run synchronously, even though they are in a different thread.