Recently it became possible to monitor Azure DSC in Azure Automation with Log Analytics. Here is how to do this setup in the Azure Portal.
First go to the Log Analytics workspace and click on Azure Resources. Select the Automation Account where DSC exists.
Select DscNodeStatus and click Save. In my example, I also want information about jobs in Azure Automation.
Verify that you have nodes added for Azure DSC and then go to Search in Log Analytics.
Do a search for Category=DscNodeStatus and you will see information from Azure DSC.
For more information and to do this in PowerShell, see the following documentation page: https://docs.microsoft.com/en-us/azure/automation/automation-dsc-diagnostics