Windows – Icon for Chrome missing after updating Windows-10 to November Update 1511

Those tiles are limited to Modern UI apps. It's possible for such an app to launch normal desktop applications, so all that's needed is to make a simple app that launches the proper desktop application and customize that app's tile.
However, making an app for each shortcut may be a bit annoying. Luckily, one app can have multiple tiles, so called Secondary Tiles. So what's needed is an app that can make custom tiles that launch different programs.

Even more luckily, someone has begun making such a program. It's called OblyTile. This will allow you to create cool looking tiles for non-Modern UI apps (desktop shortcuts).

Features:

• A tile can point to a program, file, URL etc. (with optional arguments)
• Custom background color.
• Custom icon/image.
• Custom small image.
• Custom text (or no text) on the tile.

HOW-TO

Create an PNG image that will cover the entire tile. The size should be 120x120px. This image can have transparency (then the tile's background color will show). Optionally, also create a small version of the image, 30x30px.
My icons for a Firefox icon: small - big
They are white and transparent, so a bit hard to see. On grey background:

Run the OblyTile program.
Fill inn the name of the program, browse for the .exe file (or what you want) and select the two images. Then select a background color for your tile. Example:

This can also be done through command line / batch scripts:

start ob.exe "Firefox" "C:\Program Files (x86)\Mozilla Firefox\firefox.exe" "" "fx_big.png" "fx_small.png" #f58d00 show

After doing this with a few tiles (Mail&Desktop tiles are not custom):

And the small image:

To remove a tile, just right click it and select Unpin from start

My icon are from this icon pack, but there's no need to go for the same minimalistic style as I did.

I no longer believe this answer to be correct! (August 2016)

I believe Enterprise editions do not get the 1511 update because of their type - Enterprise is the version you roll out to Employees of a company; you don't want them upgrading their Windows version (and potentially losing compatibility) without your explicit say-so as the administrator.

If you download the 1511 ISO from MSDN and mount it or burn it to a USB stick, you'll be able to run it from the Desktop and install it without losing any files. This, I think, is the only way for Enterprise users to get the update.