This behavior isn’t “strange” ... I think you hit the nail on the head as to its cause and the general remedy.   I have not yet personally used CGI::Fast, preferring instead to use Plack, but it does indeed appear to me that your FastCGI processes are crashing, after which they are necessarily being re-spawned.   A completely unhandled exception will cause any script to die, so this outcome must be prevented, because the causes of exceptions in production code cannot.   I think that the entirely-correct strategy is to wrap the body of that while loop inside of an exception-handling block, e.g. eval, so that any exception, no matter what the cause, will be explicitly trapped and some traceback given in a log.   (You may have to write code to do this.)   I am frankly puzzled why the examples for CGI::Fast do not include some kind of exception handling.   In general, the synopsis given seems simplistic, although I’m sure the code works well.

As I said, I customarily use include Plack, and, unrelated to that, part of the standard preamble for my modules is this:

use Try::Tiny;
use Exception::Class;
use Exception::Caught;
use except::classes; // a set of app-specific exception classes
use Error::Return;

You may wish to peruse some of these.   HTH...