Melt butter in a saucepan over medium-low heat. Simmer until butter starts to turn a warm amber color and gives off a nutty aroma. Watch it carefully and don't allow it to get too brown as the liquids will start to evaporate (see NOTES below). Pour the butter into a mixing bowl, cover, and let cool in the refrigerator for 15-20 minutes.
Once it is cooled, add brown sugar and sugar and cream for 4 minutes. Scrape down the bowl. Add eggs one at a time (if making the full recipe), mixing after each addition. Add vanilla and mix.
** In order for it to be a thin and crispy cookie, it is imperative to not cook the butter too long. If the butter is browned too long, some of the liquid in the butter will evaporate. This will produce less overall butter in the cookies and the cookie dough will be slightly crumbly and dry. This will create a puffy cookie. If you aren't sure how long to let the butter brown, I would suggest letting it melt and simmering for about 2 minutes after it is melted.