I have made these cookies a few times now, and they have replaced my 'normal' chocolate chip cookie recipe. The browned butter adds a depth to the flavour.
The first time I made them I burnt the butter, but no worries! It just tasted a bit like creme brulee!!! My other favourite desset!! I highly recommend them!
How to brown butter:
Put your butter in a frying pan (one without a non-stick coat works best so you can see the colour change)
Turn the heat to medium, and continuously swirl the butter in the pan over the burner.
You will see bit of it starting to turn, but how you really tell is the butter takes on a nutty aroma! It smells delish!!! As soon as you can smell that lovely smell and turn colour, take it off the heat.
Note: If it is really hot, i found that if you take it off the heat and stop swirling, it will spit hot butter! So I take it off the heat, and keep swirling it on a cool burner until it cools just a bit
I'd imagine you could add browned butter to any of your baking adventures, you would just need to add a bit more dry ingredients to even it out.
Obviously best with milk!
You can find the link to the recipe here