There's a method suggested here, based on this XDA thread for both rooted and non rooted devices:

Rooted Devices

• First Method: Download a file ( chrome-command-line) mentioned in the article, change it's properties and place in appropriate directory, relaunch Chrome

  
  



• Second Method: From the XDA thread linked, download ChromeDE.zip and flash

  
  

Since I don't have access to PC, tried the second method and it worked. Chrome now opens in default desktop mode ( Moto X Play, 6.0.1, Chrome 52.0. 2743.91)

Unrooted Devices (not tried by me)

• Download the file mentioned in first method above

  
  



• Using adb , enter

  
  

  adb push /chrome-command-line /data/local/chrome-command-line

  
  

Thanks for asking this, I have been wanting to do this but was too lazy to search

Edit: User Agent Switcher app for rooted devices can do the same, and in addition offers a host of other choices and creating shortcuts