In the root, create a file named AppProxy.cfc. Its contents are thus:
<cfcomponent output="false" extends="application" displayname="Application.cfc Proxy" hint="Extends the root application object so that subdirectories may extend it.">
</cfcomponent>
Then, in your subdirectory, set up your application.cfc to extend AppProxy.cfc. This will successfully inherit your root directory application.cfc methods.
<cfcomponent output="false" extends="AppProxy">
<cffunction name="onRequestStart" output="true">
<cfset super.onRequestStart() />
<!--- Some other stuff happens here. --->
</cffunction>
</cfcomponent>
This will work, by the way, even if the AppProxy isn't in the root directory. In that case, make sure your "child" application.cfc uses dot notation to find the AppProxy.
<cfcomponent output="false" extends="Path.To.Child.Directory.AppProxy">
<cffunction name="onRequestStart" output="true">
<cfset super.onRequestStart() />
<!--- Some other stuff happens here. --->
</cffunction>
</cfcomponent>