You mention: "we are using the Tridion ASP.NET Web application" but actually there is no such thing.
What you are doing is publishing ASPX pages, which as Nuno already mentions in his comment, will trigger a recompilation of the web app. That is what is causing this high CPU usage whenever you publish. This is because all your ASPX pages are considered a web application by IIS and that is not designed to deal with updating single ASPX pages. The reasoning why you didn't see it before is most likely related to the version of the OS and IIS, remember high CPU usage actually indicates a maximum use of resources.
To fix this issue you should not publish ASPX pages, but actually deploy a web application and have it use published content only. Specifically for that we have released DXA, see:
- https://www.sdl.com/download/data-sheet-sdl-tridion-sites-dxa/111563/
- https://appstore.sdl.com/web-content-management/list/digital-experience-accelerator/
- https://github.com/sdl/dxa-web-application-dotnet
There is a new release of DXA (version 2.0) coming up this month, so keep an eye out for that.