Hi,
Right now there is not a simple button to restart the shiny applications, its something its often requested and i added one more vote for our product management team.
It is possible to get the PID of the shiny applications if you go to the Connect Admin page. Just login with a user that has admin access to connect and click on the admin button in the top navigation.
At the bottom of that page you can see a list of all processes running that are managed by RStudio Connect with the corresponding PID. You can then kill that process without restarting Connect.
I hope this helps, let us know if you have any other questions!
Daniel