How about long running work that is done using an async method

public class QueueProcessor
{
    private readonly BlockingCollection _messageQueue = new BlockingCollection();
    
    public void StartProcessing()
    {
        //Start background processing here. What's the best approach?
    }

    public void Enqueue(Message message)
    {
        _messageQueue.Add(message);
    }
    
    private async Task ProcessQueue() //notice the async Task here
    {
        foreach (var item in _messageQueue.GetConsumingEnumerable())
        {
             await ProcessItem(item).ConfigureAwait(false);
        }
    }
    
    private async Task ProcessItem(Message message) { }
}