I was struggling with this as well. Fortunately, Steve Michelotti documented a solution that worked for me here.
At the end of the day, I enabled all verbs (verb="*") to the ExtensionlessUrlHandler-Integrated-4.0 handler in my web config.
<system.webServer>
<validation validateIntegratedModeConfiguration="false" />
<modules runAllManagedModulesForAllRequests="true" />
<handlers>
<remove name="ExtensionlessUrlHandler-Integrated-4.0" />
<add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="*" type="System.Web.Handlers.TransferRequestHandler" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0" />
</handlers>
</system.webServer>
Others have pointed out that having WebDAV enabled causes issues. Fortunately, I did not run into that issue as well.
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