In a small saucepan over medium heat, melt butter and stir frequently. Butter will begin to foam and boil. After 4 to 5 minutes you will see brown flecks appear in the bottom of the pan. Continue to stir until butter is golden and flecks are a medium brown color. Watch closely, butter will brown quickly. The entire process will take about 5 to 7 minutes.
Remove butter from heat and pour into a bowl. Refrigerate for 15-25 minutes, or until butter is just solidified. You're looking for a room temperature or slightly cooler, solid butter.
In a bowl, cream together brown butter and sugar. Add vanilla and stir to combine.
In a separate bowl, sift (or whisk) together flour and salt. Add dry ingredients to butter mixture and mix until a dough forms. Add 1 to 2 tablespoons of water as needed to bring dough together (this accounts for the water cooked out when browning the butter). Dough will be slightly crumbly, but will hold together when pinched between two fingers.
Toss dough out onto wax or parchment paper and form into two logs, about 1 ½" wide. Refrigerate for an hour (up to 24 hours).
Preheat oven to 350 degrees Fahrenheit. Line a baking sheet with parchment paper and set aside.
Remove dough from refrigerator and unwrap. Using a sharp knife, slice dough into ¼ inch slices. On a sheet of wax paper or parchment, pour a small amount of turbinado sugar. Roll the edges of each cookie in sugar and place 2 inches apart on prepared baking sheet.
Bake for 12-14 minutes or until cookies just begin to lightly brown around edges. Remove from oven and allow to cool for 2-3 minutes before transferring to a cooling rack.
Cookies will keep for 5-7 days in a sealed container. To help keep cookies fresh, place a slice of bread in the container with the cookies. Replace the bread slice as needed.
Raw cookie dough will keep in the refrigerator for 3 days. Cookie dough and baked cookies freeze well, up to 3 months in a sealed plastic container or freezer bag. Allow frozen dough to thaw for up to 10 minutes on the baking sheet before baking, or add 1-2 minutes to the baking time.