How to manage to write an application (JS + PHP) that would let the user resume the broken upload so as not to reload it? N.B. No frameworks are acceptable.
No, you cant do that simply by using JS+PHP. When submitting a form the browser submits the whole file, you cannot tell it to sumbit only parts of it.
However, you may be able to do what you want by using a combination of Flash or Java+PHP, but you have to override the uploading routine implemented by PHP...