Place the butter in a medium saucepan. Cook over medium heat, stirring frequently. Once the butter melts and begins to foam, continue to cook until brown specks begin to appear beneath the foam. The butter will have a nutty aroma. Remove from the heat and transfer to a glass bowl. Allow to cool about 10 minutes, or until warm to the touch but not hot.

Whisk together the flour, baking soda, and salt. Set aside.

Using an electric mixer on medium speed, beat the cooled butter, brown sugar, and sugar until combined. Add the egg and vanilla, and mix well.

Reduce mixer speed to low. Gradually add the flour mixture, mixing just until combined.

Refrigerate the dough for 30 minutes.

Using about a tablespoonful of dough at a time, roll into balls and place on the prepared pans. Flatten each ball slightly. Sprinkle coarse salt on the tops of cookies. (The amount of salt is a preference thing. I recommend just barely a pinch.)

Bake 12-15 minutes, or until the edges of the cookies are browned and the centers are set. Cool in pans on wire racks for 5-10 minutes. Then, transfer the cookies to wire racks to cool completely.