In a small saucepan, heat the butter over medium-high heat until melted. Keep cooking, swirling the pot occasionally, until it starts to turn brown and smell nutty. Remove it from the heat, pour it into a bowl (scrape out all the nice browned bits at the bottom) and stir in the sugars. This should cool it down somewhat, but wait a few minutes anyway before stirring in the eggs and vanilla. Then add the flour, baking soda and salt and stir with a spatula until almost blended; add the chocolate chips and stir until the dough comes together. Refrigerate for at least an hour or two, or overnight.
When you're ready to bake, preheat the oven to 350F. Scoop out walnut-sized balls of dough and space them an inch or two apart on a cookie sheet that has been sprayed with nonstick spray or lined with parchment or a silpat mat. Bake for 12-14 minutes, until golden around the edges but still soft in the middle. Transfer to a wire rack to cool.