The Windows 10 Start Screen is basically a reimagined app shortcut storage location that brings Windows 8/8.1 capabilities into the hands of Windows 7 users. But, customization is limited. For some things in Windows 10, it's easy to choose to pin tiles to the Start Screen.
What precipitated me to want to figure this out is the Facebook app for Windows. The Facebook app for Windows, introduced early in the Windows 8 days, hasn't been updated in a long time and it just plain sucks. The Facebook web page is much more useable, so why not just pin a web page shortcut to the Start Screen instead?
I expect this to improve over time with updates to Windows 10 and Microsoft Edge, but for now, here's one convoluted way to do it.
Open the web site for which you want to create a shortcut in Internet Explorer, right-click on a blank part of the page, and choose "Create Shortcut." NOTE: This DOES NOT work in Microsoft Edge, since Edge doesn't offer the ability to generate a web page shortcut (yet). The shortcut will be created on the Windows 10 Desktop (this is your only option).
Right-click on the Windows 10 Start button and choose Run.
Using the Shell command (which I've been covering a lot recently), use the following command to open the Windows 10 Programs folder: Shell:Programs
Go back to the Windows 10 Desktop, right-click on the newly created web shortcut and choose Copy.
Jump back to the open Programs window, and choose Paste to install the shortcut.
As you can see in the example, placing the web shortcut into the Programs folder, causes it to show up in the All Apps list in the Start Menu.
Locate this new web shortcut and drag it to the Start Screen.
Note that just because you created the shortcut using Internet Explorer, doesn't mean that Windows 10 will open the website using Internet Explorer. Windows 10 associates web shortcuts with your preferred web browser. By default, it opens web sites in Edge.
Know an easier way? Let me know.
BTW: As one commenter suggests, it's probably just easier to tap or click the Ellipses in Microsoft Edge and choose "Pin to Start."