Personally I don’t believe there is one question. But I do think it’s a legitimate question to ask for evidence about why we are doing something. 

However the nearest that comes to it ( arguably) is employee net promoter score. ( definitely not perfect)

“How likely are you to recommend us ( sometimes “as an employer” added) to friends and family “

Lots of articles around this. Here’s one

www.netigate.net/.../

( it may just be me but the wording in your questions seems a little American. If someone asked me about my workplace I would think they were asking about my physical place of work) 

1. How happy/satisfied are you working here? Not very= 1, 2 , 3 4, 5 = highly (Something like that anyway).
2. What would improve your happiness/satisfaction here?

Should turn up both issues and cures perhaps?

In reply to Keith:

Thanks Keith, that article is very useful.

In reply to David Perry:

Thanks David, we are going to include in our next engagement survey and see what comes up!

the COO is still adamant there is just one measure that will tell him how 'happy/satisfied' we all are

He's right. It's called the "sickness absence rate". Although I also like to match it against the "employee turnover rate" and "holiday use rate".

The problem with asking your employees how they are feeling about their work satisfaction is that *employees do not trust their employer*.

This is true of almost all employee populations, regardless of how good the relationship is and/or how committed they are to the work. Employees don't trust employers enough to consistently (as a population) tell them the truth in a usefully measurable way.

That's not to say that surveys are useless. They have value in terms of setting baselines and sometimes getting specific insights into problem areas. But they aren't good as a one-off measure of satisfaction.

Generally, by far the most reliable indicator of employee satisfaction is sickness absence. If it's below your industry standard, satisfaction is probably better than average. It's above, then the opposite. If it's below *and* you have an OSP programme, then it's very good.

If you have no OSP and your sickness absence is below average, you should check it against the turn over rate. If it's more than ~15% (adjusted for industry), people are too scared to take sickness absence and leaving instead.

If you have OSP *and* a low absence rate, check your holiday use rate. If employees are consistently taking all of their holiday entitlement, and you've got a low sickness absence rate, *and* you offer OSP, you have stellar employee satisfaction. But if few employees take all of the holiday entitlement they are allowed, then you have a culture of fear (probably got a turnover rate above 20%).

The best things about all of these figures is that you don't need a survey to get them and, because they measure behaviour not opinion, they are objectively reliable.

I'm not sure if this is the kind of thing you're looking for but Gallup have the 'science' behind their net engagement score questions here:
www.gallup.com/.../q12-question-summary.aspx
Unfortunately they don't sign post where a lot of their stats are from.

In reply to Robey:

Thanks Robey, that's really helpful.

In reply to Rachel:

Thanks Rachel