Marco Manetti has recently expanded the 22-page lecture notes from 1999 mentioned in the OP into a 574-page text book: <A HREF="https://books.google.nl/books?id=l9l-EAAAQBAJ">Lie Methods in Deformation Theory</A> (2022).   
It is advertised as *"being the first book to apply the differential graded Lie approach to deformation theory of complex manifolds".*    
I have the impression the pdf can be downloaded freely from <A HREF="https://link.springer.com/book/10.1007/978-981-19-1185-9">Springer</A> (but perhaps that capability is tied to my IP range).