Classic Start Menu cannot search network shares but using an app like Winaero Librarian, you can include a network share or mapped drive letter inside a Library and then use the "Search" shortcut inside Start -> All Programs to find content from that location, although it isn't really indexed.
See this article first:
http://winaero.com/blog/how-to-include- ... libraries/ (How to include UNC network folder paths and shares in Libraries). It applies to Windows 10 too. Once your network folder is included in any Library, open the shortcut called "Search" inside All Programs. That is the old Windows 8.1 search and it is present in Windows 10 too. It is capable of searching network shares.