I love the flavour of browned butter, and I've tried baking sable cookies with roasted flour (delicious). Is there a good reason why, in a bog-standard chocolate chip cookie recipe which calls for melted butter, I could not use browned butter? Will the fats react differently in the baking process?
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.
While brown butter works great in chocolate chip cookies, one caveat that I haven't seen mentioned in other answers is that browning butter removes the water content, which can potentially have an effect on your end bake. Not to worry, though; you can just add the lost water back once you're done browning the butter.
Kenji Lopez-Alt of The Food Lab actually uses an ice cube to simultaneously cool the butter to workable temperatures and replace the water.
See this source. It is a good read if you click over to the full write up.
If your recipe calls for softened but not melted butter, make sure to cool the butter until it is softened again so that the air bubble network from the creaming method can re-form. The creaming method doesn't work with melted butter, browned or otherwise.
Using a tip from America's Test Kitchen I've also done cookies calling for softened butter with melted browned butter and two extra egg yolks, where this makes the cookies chewier and less cakey.
Brown butter is great in cookies. I have not found from experience that it changes the way the fat reacts during baking. The browning just adds that very lovely and distinct nutty flavor profile to the cookies. The Ambitious Kitchen food blogger uses brown butter in many of her cookie recipes that call for melted butter so you could compare her ratios of fats to carbs and moisture levels to recipes that you currently use and play with it from there if you want to get really technical.