Baking – Why are the cookies crumbling after I bake them

You can absolutely do this. I do it in brown-butter cornbread, for example. A good trick to boost the flavor even farther is to add non-fat milk powder to the butter as it browns, to supply even more protein for the Maillard reactions.

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.