I don't know the direct answer and recording the act of pinning a template did nothing. Then I thought it might be doable by searching through the registry but I didn't have much luck with a quick look apart from finding things in the Most Recently Used sections. So I'm going to distract you by asking if you have considered alternatives.
It takes a user two clicks to get to the New menu (File then New) so pinning something there is not going to improve the user experience much. IMHO it would be better to create a button on the QAT or Home Ribbon Tab to launch a new document from the template you want to access.
Charles Kenyon has some useful insights here
Redirecting
Another alternative is to use Greg Maxey's addin to organise all your templates. See
File New Interface for Word 2013 - 2016