Answers, we got answers...

One of the anonymice had a nice answer for the open book test. The short version is that Essex and Co. are arguing againsta strawman of their own contrivience and turning into knots doing it.
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While it is certainly true that employing a broad spectrum of averaging methods and being inconsistent about their application can lead to a broad spectrum of results, this does not speak to what was actually done by NASA in this case.

By diving (and subdividing) into small regions (and representing the temp change within each small region with the average temp change over the region), NASA has already accounted for the "extensive vs intensive" issue that ME&B harp on.

But NASA has done much more in the process. They have compiled information about and then tracked how each region and subregion has behaved. This information is used as a cross check on what the mean global anomaly is doing.

If we tracked surface air temperature over time from every square inch of the earth's surface, few would argue that we could not then draw conclusions about whether the air at the surface of the earth was warming or cooling as a whole.

For example, if all the temperatures went up, we could conclude it was warming. If all went down, cooling.

But we could do much more because, since the temperature measurement at each location correlates directly with the average translational energy of air (N2, O2, etc) molecules in the "parcel" of air above the surface, each temp measurement would represent the same "extensive" thing (in this case translational energy).

A change in the temp of each parcel (temp anomaly) would represent the amount of translational energy lost or gained by each parcel.

The total obtained by summing all these together would represent an actual physical quantity -- the amount of energy lost or gained as a whole (ie, by all the parcels)

So dividing this total by the number of measurements (ie, simple arithmetic average) would also mean something physical (the average change in the translational energy of a parcel of air above the earth's surface).

So, the ME&B argument basically boils down to the following claim: "there is no way of using averaging to 'fill in the gaps' in order to come up with a reasonable estimate for the changes occurring in those gaps" (in this case, parcels of air above small "patches" of the earth).

They have claimed something that is clearly false. Any good engineer knows this (and regularly does it, though not necessarily for the earth's surface) -- and so does NASA.