As I said, it was "select count()" on MyISAM; this is well-documented as being dog-slow in MySQL.
Something interesting I did learn while researching was that the MySQL query cache is case-sensitive (and byte-sensitive, too), so, assuming none of the following have been issued within a set time frame, NONE of them will be served from the query_cache:
select foo from bar where baz=123;
select foo from bar where baz = 123;
select foo from bar WHERE baz=123;
SELECT foo from bar where baz=123;
and so on.
I spent some time looking through the "commercial" code one of our sites was built on; they apparent didn't know either as there were UPPER and lower-case SQL statements within the same file and even sometimes in the same method.
Chalk one up for having a rigid coding-style manual 😉
The current solution, which I dreamed up just prior to a meeting with an "outside" expert who suggested the same thing (WHEW! Glad I included that as a bullet-point in the meeting agenda!), was to have another box analyze the DB and write a summary table of counts back to the live box.