We are developing a Web application with couple of screen
Although the requirement of you client does not make much sense for me but you can handle the unhandled exception using Application_Error
event in global.asax
void Application_Error(object sender, EventArgs e)
{
//Your logging code goes here
}
How to: Handle Application-Level Errors
This code example shows how to create an error handler in the
Global.asax file that will catch all unhandled ASP.NET errors while
processing a request — in other words, all the errors that are not
caught with a Try/Catch block or in a page-level error handler. In the
example, the handler transfers control to a generic error page named
GenericErrorPage.aspx, which interprets the error and displays an
appropriate message.
As a additional Note for Desktop application like WinForms or Window Services you can using AppDomain.CurrentDomain.UnhandledException or Application.ThreadException etc.