9 out of 10 times Edge isn't capable of sticking a good picture icon while creating a Tile for a Pinned Site. A sad "e" is what you only get.

IE11 is slightly better, yet it uses the small favicon with no possibility to change that.

(Metro IE of 8.1 was terrific in that regard. It was awesome, with the chance to select among several pictures. And they were all-tile wide!

With the following we're able to change the picture in the Tile. NOTE: It won't be an all-wide tile picture though, just a smaller icon; but still:
- you can choose whatever you want (IE11 doesn't let you decide the pic)
- you can open it with Edge (or IE11 if you prefer)
- it stays in the Start Menu

Do that:

1) Pin the site with IE11, not Edge with the "Add site to apps". it will add the Site to the All Apps list
2) Just pin to the Start Menu as usual from the All App (or recently Added) list
3) Now open (with Win+R for example) this folder: "%AppData%\Microsoft\Windows\Start Menu"
(you'll find your recently pinned site(s) links)
4) with Explorer open a Shell prompt with menu File\Open a command prompt
(we need to rename the extension from .website to .url)
5) launch the command "ren NAMEOFYOURLINK.website NAMEOFYOURLINK.url" with the real name of each site of course
6) in Explorer you now can right click each link and select Properties
7) .url allows you to change icon. Choose whatever you want. I currently use some useful PNG 2 ICO website to have a lot directly from the internet. You choose it and it changes immediately in the Start Menu
8) if you want Edge to open the Tiles, leave them as .url. Job done
9) if you want IE11 to open them, change back to .website by using the command prompt. For example for legacy sites that doesn't work with Edge (requires Silverlight for example), or where you still NEED Tracking protection Lists for AD removals that IE11 provids, while Edge not (yet).

Hope it helps.
Anyone willing to screenshots and tranform the above in a Tutorial is well accepted.

PS: if any developer could find a solution and let an app to reintroduce the automation process of creation of a all-wide picture of a Pinned Site offered in the ages of Metro IE, it would be awesome. For example let the click of its created Tiles being redirected to Edge. It would be awesome. A lot of Kid's tablets are in the need of such a beauty.