Create email with existing ticket

[sid-cutshort]

Is there an API or a workaround for creating an email that is associated with a ticket and all contacts/companies associated with that ticket? Essentially, I want to do what the button from the screenshot does via API.

[dennisedson]

@sid-cutshort 

Have you explored the Engagements API?

[sid-cutshort]

@dennisedson 
Yes, I have tried that, however that was designed for interactions off Hubspot. The screenshot I provided in the original question allows me to 
1) Start an email thread 
2) The email thread is associated with all the contacts and companies associated with the ticket.
The engagements API doesn't allow any interactions (like reply). It is simply a log within the ticket for analytics.