What I have:
I used the Azure Log Stream to help figure out what was going on. As Karishma Tiwari - MSFT said, the issue ended up being my web.config file (which was built for .NET 3.5 and not 4.7)
Here's the tutorial to set up Log Streaming: https://blogs.msdn.microsoft.com/azureossds/2016/09/28/how-to-identifyreview-errors-on-php-applications-in-azure-web-apps-using-log-stream-service/
My particular app's solution was to update my Azure App Service to use .NET 3.5, in its Application Settings, like so: