Hi Jack, this took me a while … 
No slow_queries log. Sorry. Shared hosting.
… but …
A staff member of my provider (Hosteurope here in Germany) migrated my installation on a fast, local machine to check performance. He experienced the exact same result: 1min for the first launch of header builder.
He checked why. Here is what he wrote:
“Header builder is calling the database and sends it to “sleep” immidiately, for about a minute. Meanwhile, the SQL-database is waiting for data, but the script does not send any. Why the script is connecting to the database and what data is transferred I cannot see or retrieve. Additionally, the script seems to call the database only when launched the first time of a session.”
Isn’t that surprising?
Any ideas?
Thanks!
m.