Restore the "Open with Code" context menu (aka right click menu) items for the user installed version of vscode on Windows. To run, simply save the file (with the .reg file extension) and double click it.

open-with-code-user-install.reg

Windows Registry Editor Version 5.00

; To use (expand) environment variables in the registry, the value must be of 
; the type REG_EXPAND_SZ. A REG_EXPAND_SZ is represented as HEX in reg exports.


; Right click on any file

[HKEY_CURRENT_USER\Software\Classes\*\shell\vscode]
@="Open with Code"
"Icon"=hex(2):25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,2c,00,30,00,00,00
; HEX value: %USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe,0

[HKEY_CURRENT_USER\Software\Classes\*\shell\vscode\command]
@=hex(2):22,00,25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,22,00,20,00,22,00,25,00,31,00,22,00,00,00
; HEX value: "%USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe" "%1"


; Right click on a directory

[HKEY_CURRENT_USER\Software\Classes\Directory\shell\vscode]
@="Open with Code"
"Icon"=hex(2):25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,2c,00,30,00,00,00
; HEX value: %USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe,0

[HKEY_CURRENT_USER\Software\Classes\Directory\shell\vscode\command]
@=hex(2):22,00,25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,22,00,20,00,22,00,25,00,31,00,22,00,00,00
; HEX value: "%USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe" "%1"


; Right click on the background inside a directory

[HKEY_CURRENT_USER\Software\Classes\Directory\Background\shell\vscode]
@="Open with Code"
"Icon"=hex(2):25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,2c,00,30,00,00,00
; HEX value: %USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe,0

[HKEY_CURRENT_USER\Software\Classes\Directory\Background\shell\vscode\command]
@=hex(2):22,00,25,00,55,00,53,00,45,00,52,00,50,00,52,00,4f,00,46,00,49,00,4c,\
  00,45,00,25,00,5c,00,41,00,70,00,70,00,44,00,61,00,74,00,61,00,5c,00,4c,00,\
  6f,00,63,00,61,00,6c,00,5c,00,50,00,72,00,6f,00,67,00,72,00,61,00,6d,00,73,\
  00,5c,00,4d,00,69,00,63,00,72,00,6f,00,73,00,6f,00,66,00,74,00,20,00,56,00,\
  53,00,20,00,43,00,6f,00,64,00,65,00,5c,00,43,00,6f,00,64,00,65,00,2e,00,65,\
  00,78,00,65,00,22,00,20,00,22,00,25,00,56,00,22,00,00,00
; HEX value: "%USERPROFILE%\AppData\Local\Programs\Microsoft VS Code\Code.exe" "%V"

Doesn't work (anymore?_

It's still working (VS Code 1.78 - Windows 10)