Baking – How to prevent a cookie from becoming spongy like a cake

This looks to me like the symptoms of too much flour - probably way too much.

Measuring baking ingredients by volume instead of weight is always a crapshoot, although it shouldn't be that far off - my guess is that either the flour was heavily compacted, or perhaps the baker's mind wandered off and she accidentally added an extra cup.

Overmixing (i.e. by using an electric mixer) is a common problem with cookies but the symptom of that is a hard and gritty cookie that doesn't rise - not one that rises too much and has unbaked flour in the center. As far as I know, that can only happen with a huge amount of flour. Obviously, too much flour would also result in a much denser dough.

What you ended up with seems to be more like a quickbread - almost a scone - and there is indeed not an awful lot of difference between the two aside from the ratios and the creaming of butter.

Excess flour probably wouldn't result in a stickier dough, but stickiness is more due to temperature than anything else. If you're finding the dough too sticky to mold then make sure it's thoroughly chilled - that will make it easier to work with.

The author of this recipe probably happens to keep in his pantry (or more professionally speaking, dry store) just those two types of flour, and so has specified a mix to get a mid-level flour with moderate protein levels, tailored to his preferences. Given that the author is Jacques Torres, this is almost certainly a scaled down translation of a professional recipe, where that is not an uncommon practice (the very odd measurements support the idea of scaling and rounding; the weird flour measurements are probably the closest volume equivalent to a weight based scaled recipe).

Two commercial varieties (all purpose, and the less common pastry flour) also have protein levels in between cake flour and bread flour, with pastry flour being somewhere between cake flour and all purpose, typically.

If you happen to have all purpose (and you almost certainly do) I would suggest using it in the recipe for the total of both specialty flours. Very few cookie recipes are so fussy that it will actually matter very much.