Save draft (Save and Continue) email

Regarding the ability to let users save their form as a draft and return later, it brings up a form where they can enter their email to get an email reminder. Is there any way to set this to automatically send an email to the address they enter on the form, and to skip posting this email form and the notice that pops up after they click on Save as Draft? In other words, I want to not have any notice or email form pop up when they choose to Save as Draft, only to send them an email.

If this is not possible with the regular settings, is there a hook we can use in an mu-plugin to modify this behavior?